Fastest way to get from Heathrow Airport to Knightsbridge London?

Two answers say the tube, but as a Londoner who does a similar journey regularly I beg to differ :) The answer is: it depends.

In light traffic (before about 6am in the morning), a taxi will take about 25 mins to do this journey (Google agrees). The same is true late at night (after 9pm). The taxi rank is also easier to get to at Heathrow than the tube station, from almost all the terminals, saving as much as 5-10 minutes in some cases.

At other times, the tube is likely the fastest option, as you won't have to change off the Piccadilly line. Getting there on the Heathrow Express is an option but the tube changes aren't great in this case.

The taxi is of course the most expensive option. But you asked for both the fastest way and the cheaper options, which don't align with each other :)

Without a doubt, the tube will be your fastest option in this case. The Piccadilly line goes straight there.

Express train goes to Paddington, at which point you'd have to change to the tube and come 1/3 of the way back!

Overland train also goes to Paddington. And a bus could take longer, depending on time of day.

Funnily, the tube is about the cheapest option - should be less than 5 pounds with an Oyster, although prices went up this year, so not sure of exact value. Bus is the cheapest, but will take longer, and you'd likely need to go to Hammersmith first and then change.