LHR to Premier Inn

One step free way is to take the Elizabeth Line to Paddington, then transfer via the tunnel/lift/escalator direct to the Bakerloo line to Waterloo-https://www.ianvisits.co.uk/articles/paddington-stations-new-elizabeth-to-bakerloo-line-tunnel-55044/

If the flight is on time and immigration is expeditious the Elizabeth Line will get very busy on it's way into London at intermediate stations- £13.90 each

Or the Heathrow Express to Paddington then the Bakerloo line (should be a much quieter train)- £27.90 each

Personally I would take the bus to Feltham Station, then step free access to a main line train to Waterloo (but that train may also be very busy, although it is a long train and runs every 10 minutes or so, I would head to the rear of the train for more likelihood of seats). It is about the same overall journey time as the Elizabeth Line.
£12.15 or £8.75 after 9.30am.

Bus 285 (double decker) from Central Bus Station for T2/3 or 490 (Single decker) from T5 at Heathrow.

the cheapest way, and ordinarily the way I would have suggested is Piccadilly Line Tube from Heathrow (which terminal?) to Barons Court for a very short cross-platform change to the District Line Tube to Westminster (step free to surface) and walk across the very scenic Westminster Bridge.

Question though - Premier Inn County Hall (for you) or the Marriott (maybe the kids)?? While both should allow you to drop your luggage, only the Marriott is likely to allow such an early check in, and that's not a cert. Have you taken a room for tonight so that you can check right in?

My caveats are two - the first is that Westminster tube station is very close to demonstrators who may accumulate at Westminster, especially as we have a celebrity in town who attracts a LOT of attention.

The second is that landing at 7 you will be heading into London at Rush hour unless you stop for breakfast before setting off, so the trains and Elizabeth Line and tubes and buses will be crowded and getting more crowded.

A car service will get caught up in rush hour traffic.

I stayed at the PI County Hall in June and took the Heathrow Express. When I arrived at Paddington, I followed the black line on the floor to the cab ranks. I don't like hauling my bags onto the tube during rush hour.

I will probably take your advice and do the same since I don't like to drag my bags around in the rush hours either. I paid extra to get checked in at 11a.m at PI County Hall. The children are staying at Hampton Waterloo and they store their bags for them.
Thank you so much for your help!

I paid for early check-in as well. I still arrived before 11, so they stored my bags while I headed out for a walk until my room was ready.

Enjoy London! I'm excited to return for Christmas this year.
Happy Travels!

I always take the tube in from LHR. It's super easy. I don't understand why having your suitcase with you is an issue. I moved my 55 inch TV to my new apartment from Manhattan to Brooklyn in the NYC subway.

@stacie and @CathyVA - We are thinking about going to London next year and staying at the PI County Hall as well. I have researched our options for getting to the hotel. The Heathrow Express and then a black cab is what I would do. We are in our mid to late 70’s and, even though we travel light, schlepping luggage from point a to point b is not what I want to do.