Visiting London 10 days: April 23-May 9

The general advice is that anywhere within the Circle tube line or just on the south bank of the Thames will be fine. There is nowhere that is uniquely better located to see the sights - London is a huge city and the main sites are spread out right across the centre. You sensibly need to use the excellent public transport to get around. So you need to make sure that wherever you choose is not too far from a tube station.
Wanting a 3 bedroom apartment you might need to look a little further out than is ideal as there are not a lot very centrally and what there will be will be expensive.
To see where the circle line is you can download the tube map here - tfl.gov.uk.

Don't forget the Coronation!😬

What is your budget?

If you have the means, staying within Zone 1 is highly recommended. It is also convenient if you want to head back to your room to work a little in the afternoon between morning and evening sightseeing. I personally like the Bayswater area as it is a convenient tube ride or even a walk away from a lot of things I like to see/do in London. But most places within Zone 1 would work. South Kensington is pricey, but Victoria & Waterloo areas are cheaper (this is all comparative, I have to add - especially since you will be there during the Coronation). If you want to stay farther out, I have heard good things about Richmond, Hammersmith and Camden Town, but I have never stayed there myself.

Good luck and have a great trip!

I worry that you have left this a bit late, especially since the coronation takes place during your visit.
Look on Vrbo and look at Marlin serviced apartments. You may need to find a place a little further out, but convenient to a tube line.

Before placing a deposit on a property, I'd recommend that you be sure to communicate directly with the landlord to confirm the location of the flat/house/condo unit. Look at the price map on the airbnb website as well. The cheapest ones say "London" but in reality may be well outside the heart of London and require a hefty commute via train, bus and/or tube. The landlord will be able to give you specific transportation information without divulging the exact location of his property, which they generally don't want to do before you commit. One way to request information is to ask in which tube zone the property is located, as well as nearby tube stops. Also ask about light rail and bus access, but remember you'll have to factor that in when thinking about getting to the major sites. With four people, 3 bedrooms, and 2 baths and a living area, expect to pay at least $700/night, with $1000 a night not being out of the realm of the possible. londontoolkit.com is a very good resource for information about types of accommodation available and public transportation options.

I’ve thought about this a bit and you may want to be a little further out from key areas due to the coronation. I would avoid the area around Westminster but Lambeth might work.

I recommend you look at serviced apartments (akaApartment hotels or Aparthotels) by reputable established companies, to avoid the hassle and possible scams of owner rentals in London.

Here are two we have used,with locations all over London:

https://www.marlin.com/

https://nativeplaces.com/

Also try Citadines, Stylotel,, Wilde/StayCity and Cove Cannon Street ( I like this one’s location a lot).

Fraser Suites has numerous properties around London and you may find something you like.

Check out Rick's lodging section, which hopefully you have already. He has a useful mental tool of thinking of the important parts of London as resembling a bottle on its side, you should stay within that.