London Shuttles

I've never heard of a hotel in London having an airport shuttle. Traffic around London is pretty bad, so in most cases your best bet is a form of transportation that runs on rails. Many of us take the Underground (tube), which is the cheapest option. The Heathow Express is another. Come back here after you've decided where you are staying, and our London experts will lay out the options for you. Be sure to mention how many are in your travel party, whether anyone has mobility issues, and whether you are light or heavy packers.

I've never heard of it, it is not a big thing in Europe, apart from airport hotels. There are usually shuttle buses to those.
The best and fastest way from Heathrow into central London is train.

Either tube (Piccadilly Line) which with connections will get you to anywhere in central London in 50-60 minutes depending on where you are going. Cheapest option (Peak : £5.10 Monday to Friday from 0630 to 0930. Off Peak: £3.10 At all other times including public holidays). See here: https://tfl.gov.uk/fares/find-fares/tube-and-rail-fares/single-fare-finder

Or Heathrow Express non-stop to Paddington. Faster (15 minutes, but only runs every 15 minutes). £5.50 if bought in advance, £22-£25 if bought on the day. See here: https://www.heathrowexpress.com/tickets-and-offers/tickets-offers#/
Only worth the extra cost if your destination is near to Paddington. Otherwise you still need to take the tube from Paddington, and you have minimal time saving "door-to-door".

You will be wanting to use the tube while you are in London anyway, so you will need to learn how it works, fares and directions.

We go by Heathrow Express into Paddington Station. That's north of Hyde Park and a good place to find hotels. The area also has good service to numerous metro stations.
We usually stay in the Bayswater/Kings Cross area just east of Kensington Palace. It's a short taxi ride to Paddington from there.

And whether to take the Underground or the Heathrow Express will depend on the location of your hotel.

Many of the airport hotels have shuttles, but not those in central London, as public transport is usually quicker than driving.

I don’t think I’ve ever seen an airport shuttle to a city centre hotel in Europe. Of course, public transit is much better in just about all European cities whereas in many USA cities driving is the only realistic option

For Heathrow, the Piccadilly line begins at Heathrow so that’s perfect as you’re guaranteed to get a seat.

We usually stay in Paddington and catch either the tube or TfL Rail, both of which are cheap and highly reliable. You can of course get a taxi or minicab to Heathrow, but I wouldn't recommend it due to the traffic and cost. I've certainly never heard of shuttle buses from hotels.