Best tube route from Heathrow to hotel

Without looking in detail, as I recall, the change at Hammersmith is just walking to the other side of the platform a few feet. Green Park though requires a lengthy walk and some stairs. You might be able to take the new Elizabeth Line to Tottenham Court, and catch the Northern line to Waterloo.

If you look on the TFL site, there is a trip planner that should give you the details.

If you choose Westminster you’ll have to walk across Westminster Bridge to get to the County Hall PI.

If weather is nice it’s good way to snap pictures looking back at Parliament and Elizabeth’s Tower.

Do you know which Heathrow terminal you’ll be arriving at? Asking because there are different travel options.

Changing at Hammersmith is only a walk across the platform. Less than 20 ft.

Changing at Green Park could mean walking, going up or down a level, walking some more, heading uphill, heading downhill, walking some more and all underground.

This is one of those debateable journeys where there isn't a "best" answer. There are a number of possible answers all reasonably good.
TfL say, and I agree, that the County Hall PI is 11 minutes walk from Waterloo, in my experience it is about 3 or 4 minutes closer from Westminster, and simpler in directional terms, especially if jet lagged. Personally, strictly personally, I would opt for Westminster with the nice walk across the bridge.
If you opt for Waterloo, TfL again follow my feeling if I had luggage to hop on a 211 bus from Waterloo two stops, drops you 1 minute walk from the hotel.
Some folks might say that is adding confusion to a tired mind, maybe.
To Westminster by tube is the change Hammersmith route cited above
The 211 mentioned above actually runs from Hammersmith to Waterloo. So you could do tube to Hammersmith, walk upstairs to the bus terminal then a scenic bus ride which drops you effectively at the hotel door. Joining at the bus starting point means you should get a good seat upstairs for the views.
Not the fastest route maybe, possibly the "best". You get bus stop announcements on TfL so you won't miss your stop.
The Tfl route from Heathrow to Waterloo is Elizabeth Line to Bond Street (rather than Green Park), then Jubilee line to Waterloo which stops at Westminster as well.

If you are changing between the Jubilee and Piccadilly Lines it is well worth remembering the sage advice of much missed from the Forum, Emma, who gave advice based on her daily use of the system and her walking restrictions.

At Green Park, do not follow the signs from one to the other. That will take you through a rabbit warren as was described by other posters above. Follow the signs to the Way Out, go up the escalator and then when just inside the ticket barrier follow the signs to the other line, go down that escalator and - Bob's your Uncle - you're there.

Thanks, Emma.

Another alternative is Elizabeth line to Tottenham Court Road, then Northern to Waterloo. The connection here is short if you are at the front of the train. Again though the time for this and the others above are all pretty similar.

For this reason I always try to stay on the Piccadilly Line. Makes getting back to the airport so much easier, even if you don't carry much stuff.

TfL say, and I agree, that the County Hall PI is 11 minutes walk from Waterloo

I believe that the York Road exit from Waterloo underground has reopened. Can any of the locals confirm this? If so, this is only about a 5 minute walk from County Hall if you are on either the Bakerloo or Northern line. The Jubilee line is further.

I would either get off at Westminster (if taking the Piccadilly/District line option) or try the Elizabeth line to Tottenham Court Road/transfer to Northern line and exit at Waterloo as suggested by Marco. Either Waterloo or Westminster is convenient for County Hall. The only caution is that the Westminster Bridge can be packed with tourists at times and this can make it difficult to maneuver through the crowd with a suitcase.

I used to live in a flat near County Hall. When I traveled from Heathrow, I’d either do
a) Heathrow Express to Paddington and transfer to Bakerloo line to Waterloo and exit via York Rd exit. This was before Elizabeth line. I think Marco’s suggestion is better option now.
OR
b) Piccadilly line to Hammersmith and transfer to District line to Westminster and cross the bridge.

Thank you all for your advice! I put it in the trip planner like Paul suggested, and it said getting off at Westminster is the best way.

So with that, and the comments about the Green Park stop being more difficult than changing at Hammersmith, I have decided to take the tube to Westminster and walk across the bridge. I'll definitely take some pictures of Parliament and Elizabeth's tower, thanks Claudia!

I don't know yet which Heathrow terminal I'll arrive at.