Hotel for family of 4

Look into Native
https://nativeplaces.com/

Native is an aparthotel brand and locations have two bedroom apartments labeled for 4 guests.

There are some hotels with quad rooms - the best way to find them is a check on booking.com for your requirements and then maybe check back with us on what the hotels are like.

One decent hotel I know of with quads is the Darlington Hotel near Paddington. It’s not my favourite area but some people like it. The hotel itself is a bit tired but very clean and friendly.
https://www.darlingtonhotel.com/accommodation/

I think some of the Imperial Hotels in Bloomsbury have quad rooms. Great location. The hotels themselves are perfectly acceptable but a bit tired and very popular with big tour groups of schoolkids etc because of the room sizes. But not a bad budget choice and they’re all in and around Bloomsbury, which is a good area to stay.
https://www.imperialhotels.co.uk/

Quad rooms does tend to put you at the budget end of things in hotels, but if you want something a bit nicer then aparthotels is the way to go. Look at Native, as recommended above. Also the Locke properties, Marlin, maybe Citadines.

You can have 2 adults and 2 kids in a Premier Inn room. This a budget chain hotel with many locations in London and very popular on the forum. The room will be quite cramped but it’s made up as 1 king bed and 2 single beds. Perfectly ok for a short stay.

I'm a fan of Locke. In this case I would not necessarily recommend Locke, most of their properties accommodate a maximum of 2 guests in a room. BUT, two of Locke's properties have apartments that accommodate 4, that is Ember Locke in Kennsington and Buckle Street Studios by Locke in Aldgate. https://www.lockeliving.com/en

Another one to look at is SACO'S property in London they also have apartments that accommodate 4.
https://www.sacoapartments.com/en/london/holborn

There are many hotels that can accommodate a family like yours.

What you can do is go to Booking.com and put in the city, dates and request a search for two adults and two children. It will ask for the ages of the kids. It will then give you all hotels that can accomodate your family and the approximate prices.

I would then go directly to the hotel to book.

I would look for Premier Inns separatel. Then are quite a few in London. All clean, quiet but with what you will need. Comfy beds.

I agree with Helen and Frank that Premier Inns might be a good choice for you in England. They are a chain but are very clean, very comfortable and consistently good. They also have family rooms that will sleep 2 adults and 2 kids (up to age 16) and are very reasonably priced. They are in a number of locations throughout England. I've stayed in them on my trips to the UK and like them very much.

One other advantage is that if you order their breakfast, kids eat for free. https://www.premierinn.com/gb/en/home.html

how old are the kiddoes?

Thank you for all of the great recommendations. Availability is going to drive this decision! We're traveling between Christmas and New Year's so not much left. Has anyone ever stayed at the "City Sleeper" rooms section of Royal National Hotel? It's like a hotel within a hotel and it's part of the Imperial Hotels Group someone recommended in this thread. They have rooms that are perfect for 4 people. Don't know much about the Bloomsbury neighborhood though. Is it fairly central to typical sights we'll be wanting to see? Good restaurants nearby? Thank you all for the help!

Bloomsbury is a great neighbourhood. It’s pretty central but also a bit leafier and quieter than the dead centre. The British Museum is in Bloomsbury, and it’s very close to many of the places you’ll want to see. Great transport links too. It’s one of the most recommended neighbourhoods in this forum.

It was me who mentioned the Imperial Hotels group with what I’d say was a cautious recommendation. I don’t know those specific rooms, and do check whether there’s aircon (if that’s important to you). But they’re generally respectable no-frills hotels.

I stayed in Bloomsbury on my first two visits to London, and really enjoyed the area. As Golden Girl said, it has a very residential feel to it, with tree-lined streets, and lots of lovely houses and hotels, but it is also close to a lot of the sights. I think you would like it a lot.

For family-friendly rooms in London, check Premier Inn and Novotel, which offer family rooms with a double bed plus singles or sofa beds.

So the City Sleeper looked really great, but we found another hotel that we all thought was a more unique property and in a great area-Lancaster Gate. It's called The Columbia. https://thecolumbia.co.uk
I saw it mentioned in this forum in a thread about triples. They have family rooms with 2 double beds and quad rooms with 4 separate beds--2 side by side to make up each of the 2 "bigger beds"). We chose the quad and I will report back after our stay! Next up, similar lodging for Bruges and Paris. Thanks for all of the suggestions!