How about a palace guard event. People dress up like a palace guard and then the watchers try and get them to smile or react.
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And the men are all cross-dressers.
We are toying with the idea of trying to get the site security dressed up in palace guard outfits. I hope they’re up for it 
We already regularly play a game which is very like the stiff-upper-lip/palace guard try to make someone smile, or otherwise react. It tends to go… quite far. I do like the monocle as prize idea though.
We could probably do a silly hat making workshop, with a dedicated Ascot style “royal enclosure” with strictly enforced ludicrous dress code for a show or something. Ooh. I like that idea.
For your monocles, my I suggest Weinersmith’s Monocles’ Gentleman’s Single Use Monocle?
Explanitory video here
You could probably find a way to combine this with hobby horses 
Almost all of the best popular music ever recorded.
Being incredible lovers.
Well alright, the first one.
A common stereotype in the USA is that the British have bad teeth.
How about doing something involving Cockney rhyming slang? Maybe making up new additions to the lexicon of “apples and pairs”, “Bristol cities”, etc.
Half the attendees say, “By Jove,” constantly, and the other half say, “I say,” while the other half say, “My word.”
Three halves? My word!
You could stage a traumatic incident, and afterwards everyone could have a nice cup of tea and a sit down.
Crikey!
On a crowded tram platform full of Englishmen in bowlers with umbrellas, one Englishman accidentally treads on another one’s toe. The one who got his toe trodden on says: “Sorry!”
Quite.
Have a game show type set up: English Food or not. Guests have to guess if Rumpy Treacle bits is a real food or not.
Maybe a tea contest like we have chili cook offs here in Texas. Who can have the most perfect tea? This would be by people hosting event and guests vote. Points to be awarded to best teacup, best style, etc. There could be a drawing for those that participated in voting. Prize? Tea!!
A photobooth set up. A backdrop of maybe a rainy countryside? Or the Union Jack. A table filled with all things mentioned in above posts: a tiara, a bowler, monocle, punk rock wig, etc. If money is tight, guests can take selfies there.
Calling people “luv.”
See how long it takes to translate this saying: Robert’s your father’s brother.
(Thanks, silenus!)
Or you could just play a game of Clue and have a cuppa. (j/k, sounds like a fun event)
How about a dog sled team where all the dogs are corgis
How about looking at one of the big agricultural shows, such as the great Yorkshire Show - there are many others. Just about as English as you can get.
Also you might try out a Bank Holiday at a seaside resort - pretty much Brit stereotype central
See if you can get a hen party from Newcastle down for the weekend…
[my bolding]
And lousy goalkeepers…