Re- travel to York.
The strike on the 15th is the tube. This means that the Elizabeth line will be running, if busier than normal, as is HEX.
You can take HEX or Elizabeth line to Paddington, then catch a bus service 205 (destination Bow Church) straight to Kings Cross, or Elizabeth Line to Farringdon then Bus 63 to Kings Cross. If you prefer there is a taxi rank at Paddington, or flag a taxi down at Farringdon, although KX is walkable from Farringdon.
All the press reports are that the Border Force strikes have been having limited impact.
It may be worth noting that National Express have a coach departing from Heathrow at 1655, change at Victoria 1750 to 1830, arrive York at 2355. Fares as of tonight are just £13.20 (which allows one change for £5) or £46 for fully flexible and refundable.
I know that's a far longer day than you would have hoped for after a TA flight, but it's a solid alternative to consider, which we know now will run.
LNER have not yet published their strike day timetables but would expect to operate a full timetable on the 15th, and most of a timetable on the 17th. It'll probably be towards the end of next week that timetables are confirmed.
You don't say if you are catching the through York to Inverness on the 17th or travelling earlier, changing onto Scotrail at Edinburgh.
But in a very worst case on the 17th it's easy enough to get from York to Leeds, from where there is a coach to Edinburgh at 1215 (arrives at 1720), from whence Scotrail run a train to Inverness.
So you will certainly get to where you want to be, just maybe not at the times you had planned.
