I don’t know about the eggs, but $3+ for a fancy donut is pretty standard in NY at places that aren’t Dunkin Donuts.
That even includes Burford Browns for £3.20 for a half dozen.
But it’s not a normal price at all.
There are different kinds of outdoor markets. The normal ones tend to be very cheap and also don’t usually sell eggs. The farmer’s markets always have egg stalls and are really expensive. Pretty obvious which one a TV show would go to.
(And Londoners do say paaand without being a comedy Cockney faking for the cameras).
Haven’t been in to London lately I see. 
London has had an explosion of ‘artisan’ doughnut shops over the last couple of years, places like
Doughnut time (actually an Australian chain)
Crosstown doughnuts
Where you won’t get much change from a fiver even for their ‘basic’ doughnuts.
As to the OP, entirely believable. London is an unbelievably expensive place to live. As soon as you step beyond the basics of existence for food, you’re paying a premium.
I like getting out of the city. I recall visiting family up in Newcastle, and we brought a box of 6 artisan cupcakes from a local baker, which cost us about 6 pounds. In London you might have got two similar treats for that price.
Also bear in mind that this wasn’t just any old outdoor market, this was, presumably, a pretty famous market in the centre of London (I’m putting bets on Borough Market, which basically has a permanent film crew) - this is probably the priciest place to buy eggs in the UK!
I buy my eggs from my local farmer’s market for about £1.75.
Harry and Paul also capture the high end food shop, very nicely
Food prices at supermarkets in London are the same as anywhere else in the country. You don’t just get the basics of existence there. There are artisan bakeries (like Rinkoff’s, for example) where £6 for six cupcakes would be completely normal. There’s a really wide variety, basically.
Good username/post combo btw 