I'm also planning a trip to the UK, but to visit home rather than as a tourist - yet I'm also stuck on what to pack... COVID has made me forget what it's like to travel, lol!
I'm guessing if you consider NYC the same as London, then your main activities will likely be shopping & city sightseeing, right? My absolute top item for blending in would therefore be an opaque pair of black leggings or tights - people wear them with everything, from the ubiquitous hooded khaki parka to ditsy floral sundresses
As for the hotel, flat shoes should be OK (we aren't an overly dressy nation) but just make sure they have good grip - the staircases leading up to those places are often no joke! In fact, that's pertinent advice for any footwear you bring with you, trainers or not - cobblestoned alleys & wet concrete after rain are also common slipping hazards!
We do indeed wear trainers (or sneakers, as you call them) in the UK, but it's what you pair it *with* them will tip us off as to whether you're a tourist or not - such as white tube socks, a bum bag ('fanny pack' lol) & baggy shorts. Unless you're younger folk who can pull off 'normcore' chunky 'dad' trainers with aplomb If you're unsure, you're likely better off sticking to simpler slip-on style plimsolls than anything with multicoloured shoelaces. A cohesive colour scheme can also do a lot to make up for casual items (see previously mentioned khaki parkas: worn with black leggings, brown boots, a cream jumper, light wash denim shorts/ chambray shirt & a burgundy scarf, you'd fit right in!).
Another vote for practical outerwear too - my raincoat gets lots of use back home, even if just tied around my waist because it's suddenly too hot to wear it in the middle of the day I'd therefore advise wearing layers - @Cardiff's right that cardigans are popular for a reason