Where in England are you going? Things like ale, pastries and cakes, cheese etc. are pretty regional.
In Paris, get a croque monsieur from one of the street carts. It’s an open face sammich with good bread, good mustard, country ham, and gruyere cheese broiled over it all. Although it’s open-face, it’s street food, and easy to eat with one hand. They’re fantastic.
England: Bubble and Squeak, Jellied Eels, Whelks
France: McDonald’s (I hate the French)
Don’t forget the ciders if you’re in Somerset. Or the Vosges.
Where’s your sense of fun? 
I want to support the idea of eating some curries in England. But I’d suggest aiming for a neighbourhood where most of the locals are from South Asia. When I visited my mother’s home town (Leicester) a few years ago, the hotel was in a neighbourhood where all the shops were for the local Indian population – sari shops, jewelry shops, etc. And when you looked at the menus for the restaurants in that street, it eventually dawned on you that they were all vegetarian Gujarati restaurants – except for the fish-and-chip shop, but that sold curries and kebabs as well as fish and chips.
If you go to Somerset, you’ll have to try real Cheddar cheese, too.
Where are the sick bags?
I see most of the posters have talked about what to get in England, so here are some suggestions for France:
Breakfast: pain au chocolat and a cafe au lait. American coffee was pretty much dead to me for life after that last one. If you want to feel like a real local, have them standing up at a counter in the Gare de l’Est railway station
Lunch: the aforementioned Croque Monsieur, or Croque Madame, which includes a fried egg, OR find a steak frites somewhere (lots of Dijon mustard, please); OR a couscous from some Tunisian-run place or another.
Dinner: try some regional items like moules frites (steamed mussels with fries on the side in the Belgian fashion); choucroute (sauerkraut with sausage, ham and pork, from Alsace); cassoulet (wonderful bean stew from the southwest). If you’re still jonesing for steak, steak au poivre or steak bernaise will get you where you need to go.
Finish off with cheese; try as many types as you can get away with. Have a digestif to settle everything down; I prefer Calvados, the apple brandy from Normandy, myself.
There are thousands of decent places to eat in Paris, but if I were to go to one place only that summons up the whole Belle Epoque feel (and with damn good alsatian-style food and impeccable service), I’d say Brasserie Lipp on Boulevard Ste. Germain.
Damn, now I’m starving.
Are you mad? In England: eat food you brought yourself, or maybe try and find a French restaurant or any place where they import food from the continent. In France: enjoy - everything will be tasty as can be! 
Word of warning - like the currency, you need to convert steak doneness. When ordering steak in France, please refer to the following conversion scale:
Well done: medium rare (and the chef may come out of the kitchen physically assault you for being such a philistine)
Medium-well: rare
Medium: really very rare
Medium-rare: really very rare indeed
Rare: roadkill
Blue: cold and still mooing
(For the genuine grades, see here.)
The UK steak conversion scale is
Well done: Desert Storm aftermath
Medium-well: RAF boot leather
Medium: Nerf ball
Medium-rare: Crocs
Rare: well done
Blue: medium-well
(Actually, there are many fine steak restaurants in the UK. The above might, however have been found in the past in chains like Berni Inns and Beefeater.)
Another geniune tip: avoid, at all cost, any pub bearing the corporate imprint of JD Wetherspoons, of which there are many. Though they do good drinks promotions and cheap food-and-pint deals, the food is often crap and the atmosphere soulless.
I don’t know if they are still around, but avoid at all costs any branches of the notorious Angus Steakhouse chain. They are over-priced rip-off merchants only frequented by unknowing tourists.
Berni Inn/Beefeater steaks are good only as drawing materials. At anywhere decent (there’s a superb steak restaurant in Borough Market…) it’s probably much closer to the French chart. I have never eaten steak in the US so I have no basis for comparison, but I would send back a blue steak that didn’t try to eat the salad it was served with.
I second Eccles Cakes.
Actually, go into a cake shop and get them to do you a mixed box of stuff. Go nuts. This will give you diabetes, but it will be enjoyable. Do you guys have vanilla slices? Get a vanilla slice in. Ooooh, vanilla slices.
Chip shops, also yes. If you’re in the North, have battered cod (actually, no - whichever fish is ethically sustainable this week. I can’t keep up) and a chip barm. Do not ask for the chip barm in the South as they will have you disappeared and made into hats. You must also try mushy peas, they’re appalling.
Try and find a proper real ale pub. Drink some real ale. In the South they perpetrate something terrible with mainly apples called Scrumpy. How many days abroad have you got?
In France, you must have a really offensive Camembert. If you drink, have Pastis. If you don’t, have citron pressé. The Camembert should hit you in the face like a hot nappy and the pressé should take the enamel off your teeth. Oh, if you’re on the coast you must have a bucket of moules. Moules are why the French have bidets.
In France, order a salad. In England, do not order a salad. I don’t know about the salads in your native area, but on the continent I spend a lot of time vibrating in herbivorous glee - beautiful things, they are, with flavoursome leaves and seeds and ripe tomatoes and olives and bits of cheese and sexy dressing and personality, as opposed to the limp English things with a quarter of an anaemic tomato, half a depressive lettuce, dessicated cucumber included with great suspicion, and a powdery boiled egg.
Actually, if you’re interested in experiences rather than necessarily good food, have an English salad. With the same justification you should also have ten pints of beer and a kebab, a Pot Noodle and a Ginsters pasty.
I also second having a wander around supermarkets in both countries. And actually, yes - someone upthread mentioned couscous. I’m pathetically addicted to French supermarket couscous (it’s usually labelled tabouleh). And celeriac rémoulade in a tub! The French probably raise an eyebrow in horror at that, but I could eat it with a spade.
If you’re in a French seaside town like Calais or Boulogne-sur-mer, try the Bouillabaisse. Delicious!
In England, I heartily recommend Tea (or Coffee) and Scones (AKA Devonshire Tea), especially somewhere like Hampton Court Palace or a proper Tearooms. Make sure they used Clotted Cream on the scones, not whipped cream.
I live in the southwest, and I’m constantly trying to find a good Cornish pasty. For example, in one of my forays I found a tiny shop in the middle of nowhere, where they boasted that their pasties were famous, and made to an ancient recipe by their 200 year old grandmother (you get the picture), but it was dried out, with chewy meat. In theory, they ought to be good, but I’m beginning to despair of finding a really good one. While by no means great, Ginsters is one of the least bad.
If you are in Cornwall, try local ice cream. Roskilly’s gooseberry ice cream is amazing.
Sausages? Why? Is there something unsanitary about the way they’re made or prepared in England?
At the recommendation of a friend who knows Paris well, my wife and I had dinner at Chez Georges. This is an old school bistro/brasserie, where you sit at tables along banquettes with no more than 18 inches separating you from the next table. Thus you see exactly what your neighbors are eating. It’s a very convivial atmosphere. When the appetizer for the table to my right came out, I asked the gentleman next to me what it was. He was very gracious. His English was much better than my French, and he said to me “it is, how you say, the face of a cow. Please - try some.” I hesitated for a second, then thought, what the hell - I’m in Paris!" It was absolutely delicious! I ordered a plate of cow face for myself.
It turned out that what I eating were veal cheeks. I don’t know if they are excellent everywhere, or just at Chez Georges, but this is the sort of thing that you’ve probably never had before that are a staple of French cuisine.
Yes, definitely eat Indian curries in England. YUM. I want to go back there again just for that. I remember that at least one or two places I ate at had rich curries made at least partially with coconut, which was delicious and which I can’t find in the States. Also I second (third?) Jaffa cakes. Also, random pastry stores (in both England and France).
Specific English foods: I adore Yorkshire pudding (kind of like fried dough, only baked, and savory and not sweet… maybe you can get this in the US, it’s in my cookbook, but I’ve never seen it) and summer pudding (a creamy berry dessert) and trifle (a creamy fruity dessert). In general desserts in England tend to be more creamy and less cakey, which I loved and which did horrible things to my cholesterol level while I was there.
If you will be in Brittany at all, you should definitely have crepes/galettes. (I think just about every other restaurant sells them, so it should be hard to miss.)
The greatest pudding of all time: Eton Mess. Strawberries, meringues and cream, all mushed up together. I would be happy to meet my maker by diving into a vat of it. (And to tie France in to this, it’s even better when accompanied by a glass of fine Sauternes.)
wolf-alice, yours is one of the funniest posts I’ve ever read. Thank you!
To the French cheese list you MUST add Pont-l’Évêcque. Divine. Stinks. Ambrosia. In the UK, Stinking Bishop comes close to this in divinity, but not quite.