Hotels in London and Airport Travel Time

There are two Premier Inns in the Paddington area. Premier Inn Paddington Basin, and Premier Inn Paddington (Station). I don't think they're as large as the PI Country Hall, but they're both good, relatively inexpensive and have the same reputation of being clean and comfortable as the County Hall one. The Premier Inn Paddington Station is the newer Premier Inn (it just opened last year) but they're both pretty near each other, so either would work for you. I would check the prices at both and see what you can find.

You might want to check into the price of the Premier Plus rooms at both those locations, which give you a little bit extra space, a small fridge, an espresso machine, and updated Premier wifi. Also, keep in mind that if you're taking two boys, you can still get everyone in one room, but you will need to make sure and mention that you have two children with you, as they will probably bring in a bed or beds into the room.

EDIT: actually, I should have told you to look for a family room, as that would give you more space in the extra beds. When you go to the website, if you scroll down to the types of rooms they have (double, twin and family), look for the family one is out of where they will bring in extra beds for you. But I think you can still get Premier Plus there. Although I'm not sure about that. It would be worth checking, though.

Have researched the Premier Inn at Paddington Basin?

Then right across the road from the Station is the paddington Premire Inn Whole area by Paddington has been built up over the last few years.

I always take the HEX simply because I’ve flown an overnight flight from LAX and don’t want to dawdle.

Also at least for my arrival day I stay at 63 Bayswater. Solo traveler. 15 minute stroll from the station to 63. Being staying at 63 for over 25 years. Room is small but well laid out.

The PI Paddington looks great though.

Although County Hall is less easy to get to from the airport, it is better placed for sightseeing. You are within walking distance of a number of the main tourist sites. I can’t think of any tourist attractions within walking distance of Paddington.

Also, easy to get to Hampton Court Palace from Waterloo station which is very near County Hall. Although you don’t have much time in London, I would consider taking most of one of the days going to HCP.

It looks like it takes about an hour to get there by taxi or public transport that would require a transfer.

I just did a similar trip in reverse - I took the Bakerloo line from the Lambeth North tube station [which is one stop beyond Waterloo] to LHR via the Elizabeth line. At Paddington there is a dedicated walkway/tunnel from Bakerloo to the Elizabeth station - it is "step free" using escalators or elevators. So from LHR you would get on the Elizabeth line, take it to Paddington, walk to Bakerloo using the tunnel, take it in direction of Elephant & Castle, get off at Waterloo, and walk approx. 10 mins to County Hall. Much cheaper than the HEX, and the Elizabeth line cars are quite nice.

My daughters and I (landed at Heathrow) took the Elizabeth Line to the Tottenham Court Rd tube stop, then changed to the Northern line. Got off at Waterloo. The York Rd Waterloo entrance/exit is very close to the Premier Inn County Hall. It was pretty easy.

I didn't stay in Paddington but can highly recommend the Premier Inn London Waterloo (literally 2 minutes walk behind Premier Inn County Hall but it's the cheaper of the two) as the location was phenomenal - it's equally proximate to Westminster and Waterloo stations so we found it connected us to much of the city easily without needing to transfer trains/buses.

For me personally, I'd recommend basing accommodation on what's more convenient while exploring London than going to/from the airport. With that said, I travelled from Australia and absolutely did not want to navigate keeping my possessions secure and wheeling my luggage on public transport while jetlagged so caught a taxi.

I have stayed at both Premier Inn Waterloo and Premier Inn Paddington Station. Both were fine, basic hotels.
Paddington Station is right across the street from the station so is very easy to get to by either Heathrow Express or Elizabeth Line. Access to tube is easy.
Premier Inn Waterloo is ideally located on South Bank for walking to many attractions or connecting at Waterloo station for train or tube.

If I were going to Premier Inn County Hall or Premier Inn Waterloo, and I was worried about time and tired children, I'd buy advance tickets on the Heathrow Express and then just pay for a taxi to the hotel from Paddington. Sometimes getting somewhere with kids and luggage easily is worth the extra expense.

Whether you stay near Paddington or not, keep in mind that if (as is often the case) you have an early morning arrival at Heathrow, you may not be able to check into your hotel until afternoon, which would detract from the advantage of getting to the hotel very rapidly (although almost certainly you could at least leave your luggage there until check in time).