Heathrow to Burnham Market

Piccadilly Line (tube) to King’s Cross St Pancras. Train from King’s Cross to Kings Lynn. Taxi from there.

Tricky one. The nearest station is Kings Lynn, 25 miles away.
From Heathrow take train to Paddington, tube to Kings Cross, then train. Journey time c. 3 hours - see Trainline for options.
Then it's taxi or bus (hour and a half).

https://burnhammarket.co.uk/getting-here/

While this can be purchased as a through fare from Heathrow don't do that as the fare is £76.70.

Buy it as rail fare into London, then a Kings Cross to Kings Lynn Single-
Monday to Friday an Anytime Day Single is £47.70
Saturday and Sunday a Super Off Peak Single is £36.60

All fares are before any relevant National Railcard Discount.

There is a way to get the Monday to Friday fare down further if you use the Elizabeth Line from Heathrow, involving the use of a Day Travelcard, and Boundary Fares- after 0930 a Zone 1 to 6 Travelcard is £16.60, then buy a Boundary Zone 6 to Kings Lynn Anytime Day Single at £41.90- both fares get railcard discount- but that may be too complicated.

How many of you are there, and where are you going after Burnham Market?

If you're not in a hurry and don't have vast amounts of luggage, the #36 Coastliner bus from Kings Lynn bus station, a short walk from the train station, runs through Burnham Market. It's takes 1½ hours but costs only £3pp under the current national capped bus fare scheme. I have to say though that Burnham Market isn't a very convenient place from which to tour Norfolk without a car, if that's your plan