We are arriving at Terminal 4 mid-day on a Sunday and staying at a hotel near South Kensington tube station. Can we take Heathrow Express to terminal 1 to get on Picadilly line or do we take the shuttlebus to Hatton Cross and board there? I understand that S. Kensington station has lots of steps to climb up so this wouldn't work for us. What about getting off Picadilly at Earl's Court (less steps?) and taxiing to hotel? Thanks for any advice.

S. Kensington tube station does have lots of steps. Have you considered
www.justairports.com? I used them on my last trip after all the recommendations here & they were very reasonable & it's so much easier especially after a long flight. It's cheaper if you pay them in cash.

Here is a quote from the LHR Web site:

"Underground passengers travelling to Heathrow Terminal 4 should change at Hatton Cross underground station to a replacement bus service. This will run every five minutes in each direction, and will extend journey times by five to ten minutes.

"Extra staff and porters will be on hand at both Hatton Cross and Terminal 4 to help those passengers who require assistance.

"Piccadilly line services will continue to run direct between central London and Heathrow's central station (Terminals 1, 2 and 3), with a slight reduction in journey times."

According to the Journey Planner at tfl.gov.uk, the South Kensington station has five escalators and there is a taxi rank at the station. Earl's Court has four escalators and eight lifts but no taxi rank.

Here's a second plus for justairports. Me and my 3 daughters used them in June twice- coming from Heathrow to Kensington and then to Stansted Airport, very prompt and reasonable. Good luck

Another vote for justairports. We´ve used them twice: from Heathrow to Hotel (mArble Arch) and from Hotel to Dover at very reasonable prices.

You have to go to Hatton Cross and it's not that bad. The shuttle buses come every minute or so and the ride is short. No worries. I was there in April

I'd recommend car services directly from the airport esp if there are more than two of you, so long as it's mid-day Sunday. M25 motorway can get crowded Sunday late pm/evening with traffic coming back to London.
Alternatively you CAN take Heathrow Express from T4 to Paddington, catch a taxi from there. But this would probably more expensive with HEX GBP26.00 round trip per person, cab probably around GBP15.