There is no hotel shuttle to Heathrow. There are convenient transportation options. Taxi/Private Car (most expensive), tube to Heathrow Express at Paddington Station (middle cost if use advance purchase far enough in advance) Elizabeth Line or Tube (least expensive).
I suggest Premier Inns are exactly the type of accommodation that you are looking for. They are also my "go to" when I head to London. My favorite location is Premier Inn Waterloo with Premier Inn County Hall (located just a few blocks away) as my second choice. These are near a major transportation hub (Waterloo) where you can get the tube, train, or bus easily. They are across the river from Westminster and that is an easy 5 minute walk.
Hints: Put the CityMapper App on your phone. This gives you real time transit solutions. You put in where you want to go, it figures out where you are, and gives you the best choices with map and real time answers. It might say walk 2 blocks and then give you a map to follow and then show you the bus stop or tube station you need and tell you when the next one will arrive. It will track your path in real time and tell you when to get off.
You tube is your friend: Use it to watch videos about "how to" use the tube, various sightseeing locations in London, etc.
Be careful of "overscheduling." I would not schedule more than 2 major attractions a day, possibly 3 if you are headed out to a show or something. Take time to enjoy and don't think that you have to see "everything." You can't and you will drive yourself crazy if you try to do that. Prioritize what is important to you and focus in on those sites.
Very little actual cash is needed in London. Don't load up on a lot of pounds because you will still have them at the end of your trip! You should plan on having a contactless credit or debit card with no foreign transaction fees. Apple Pay is another option.
Have fun!