No, London is not formal. Be comfortable and enjoy yourselves; you are most unlikely to stand out.

If you don't wear tennis shoes/sneakers you will be doing yourselves a favour. There is nothing that advertises "American tourist" quite so loudly as sneakers, and you will be treated accordingly, which can be with avarice/contempt/disgust depending on the person you are dealing with. (The next loudest advertisement is fanny packs, followed closely by the American flag - if you look you will see few Brits in London wearing or displaying anything that looks like the UK flag.)

Other than that, you will get away with smart casual - casual trousers, polo shirts etc. - for just about everything except a handful of the very smartest restaurants which will expect a jacket and tie (and even then not necessarily a suit). If it is the Sheraton Park Lane Hotel on Piccadilly that you are talking about, I would be astonished if the tea room fell into that category.

There are one or two entertainment venues where you may also feel a little uncomfortable if you are not wearing a jacket-and-tie standard of dress, but these are very specialised places like the most expensive seats at the Royal Opera House. Even there there is nothing that obliges you to dress up and with a sufficiently thick skin you might never notice anything.