There are not any resort fees. There are taxes from the city which is out of Chelsea savoy or any other hotels control.
Not that I am aware of.
We didn't pay anything extra. We booked through Expedia & paid for flight (with Virgin ) & hotel, didn't notice anything else to pay at the time. We certainly didn't pay the hotel anything while we were there.
Get an answer from Chelsea Savoy Hotel staff and past guests.