Fastest transportation from Heathrow to Buckingham Palace

My guess would be Piccadilly line direct from Heathrow to Green Park and then walk through Green Park itself to the Palace, about 5 minutes without baggage. Green Park is the nearest underground (subway) station to the Palace and buses don’t run in front of it either.

Only the Piccadilly, Jubilee and Victoria lines go to Green Park but none of these have interchanges with the Elizabeth Line (nor go to Paddington). You’d have to get the Bakerloo Line from Paddington to Oxford Circus and change to the Victoria Line there. Or, Bakerloo to Baker St and Jubilee from there.

With two changes, I can’t see this being much quicker than the Piccadilly line direct to Green Park.

Rather than traipse around Green Park with all your luggage, why not take the Picadilly line all the way to St Pancras first and store your bags at one of the storage outfits, then back to Green Park hands free?

I bet the King has a helicopter :)

"Only the Piccadilly, Jubilee and Victoria lines go to Green Park but none of these have interchanges with the Elizabeth Line" - this is not quite correct- you can transfer from the Elizabeth Line to the Jubilee line at Bond Street. That would actually be my preferred way to do this journey as I don't love spending an hour plus on the Piccadilly line- so I would take Elizabeth Line to Bond Street, then transfer to the Jubilee line one stop to Green Park. I don't think you'd be able to take any big bags into the palace or any related buildings (e.g. Royal Mews, King's Gallery), however, so unless you're just planning to see the outside you need to think about what you plan to do with your luggage.

National Express bus to Victoria.

Walk round from Victoria Coach Station Arrivals Hall to Buckingham Palace, leaving your bags at Excess Baggage at Victoria Railway Station, then Victoria Line afterwards to Kings Cross/St Pancras.

Or Piccadilly then District line to Victoria and as above.