Avoid jellied eel. Cockles and winkles are edible, but not for the faint of heart. Pub grub is great, except for the mushy peas. Why do they do that to such a fine vegetable anyway? My wife and I haven't been to London in an awfully long time. It's a great place, and it's where she grew up.
At the risk of derailing the thread for a minute, I'll chime in here (since London is my hometown and food is a passion of mine!).
Mushy peas are (a) traditionally more of a dish from the North of England, and (b) made from dried peas, not fresh.
Pub grub is certainly the way to go. Please note that London also has a very large population of Indians, so try their food too. It's much more traditional there, and not Americanized as it tends to be here. Careful, though. You may see a dish you know that you usually order 'mild' or 'medium' but to them it's just that dish and is as hot or mild as the dish needs to be to be authentic. Don't order the Madras and hope for mild...
No real point in having jellied eels (I'm not fond of them, but I do like stewed eels!), cockles, winkles (I am very fond of them) etc. unless you're curious.
If you're on a tight budget, I'd suggest looking for the cafés where the construction workers go. They won't eat small portions, they won't go anywhere dirty, won't eat bad food, and won't pay too much. Follow them. The cafés are more traditional English fare. Remember there are not going to be bottomless drinks. You want another? You buy another.
If you're looking for a fish and chip shop, avoid anything that has tons of Union Jacks everywhere. They're tourist traps, and you won't get the real thing. A lot of good ones are just named 'fish bar' not "Fish'n'Chips.' May or may not have a table or two. Not really sit-down restaurants. They probably won't have mushy peas.
For accommodations, try something not in the central area. They'll be much more than anywhere else. The public transport is so very well planned in London that you can be in the center from the outskirts in under half an hour most of the time.
Not sure if Camden market is back up after the recent fire, but that area is a hoot on a Sunday morning/afternoon.
The Thames festival is in September, so get to the river (either side) to catch lots of events. Suggest the South Bank area as a good starting point.
