OP I'm bemused by your belief that anywhere in London you'd have to be concerned with dress.
Google "London Crowds," and click on images. You'll see Christmas shopping crowds pictures.....yes those were taken in cold weather so coats and sweaters but how many in the photo look posh????? Honestly, jeans are a norm or for Spring nice capris or light weight slacks. So unless you are invited to dinner with the Queen casual dress is what you will see throughout London. Flip flops wouldn't be wise though for a variety of reasons.
Secondly London is a very walkable city. Using Google Map's satellite overhead view, type in Big Ben. Note how close Buckingham Palace is to Parliament Square. Close by are Westminster Abby, The Thames Rive Path, London Eye, Churchill War Rooms, Horse Guards, Trafalgar Square, Leicester Square, Covent Garden. All easy strolls.
As far as hotels Premiere Inns are a good price and scattered throughout London. The Luna and Simone suggestion is a good one. The Jesmond Dene is also often recommended on this forum. Of course, read Trip Advisor reviews.
If you want a neighborhood-y investigate the Marriot Regent's Park. Hotel is a 5-7 minute walk from the Swiss Cottage underground station. Easy 10-20 minute tube ride to Westminster and the aforementioned Parliament Square.
From the hotel it's a 15 minute walk to Regents park through some tree lined residential streets. Even Less time to Primrose Hill and a nice POV of London not many tourists venture to see. Swiss Cottage has undergone gentrification in the last decade. Starbucks, Costa Coffee, Marks and Spencer Eats, KFC and Mickey D's line the nearby Finchley Road. I enjoyed sipping coffee in the teeny cafe in the Swiss Cottage Library but I'm uncertain it is still operational.
I've been to London many times so didn't need to be in the center of London so chose this Marriot as I got a Priceline deal which couldn't be beat. $99.00. Room had 2 double beds, flat screen TV, and sat in the back of the building. Extremely quiet. Good staff.
On a more recent trip met up with friends who through either HomeAway or AirBnb ( I forget which) had rented a studio apartment near Tower Bridge in the Butler's Wharf complex. That area has been transformed and revitalized as well and again easy walking to the Tower of London via Tower Bridge or stroll along the Thames Path past the Hayes Galleria (this area is very modern) and up and over to the Shard.
Truly depends on what you wish to spend and what is your idea of “neighborhoody.”
If you were to stay at a Premiere Inn Euston St Pancras or the Jesmond Dene you'd see what I'm talking about as you meander over to the newer Coal Drop Yards, then stroll along the Regents canal. Or stroll behind the Premiere Inn Euston and walk around Cartright Gardens then onto the Lord John Russell pub for a pint. Then out the door walking along Marchmont to Tavistock place, turn right and walk over to the Petrie Museum of Egyptian Archeology. A stellar non touristy museum.
Enjoy your May visit. Great city to explore.