train from Heathrow to York?

You don't HAVE to buy your train tickets in advance. If you look on the National Rail website, you'll see that there are trains leaving at least every half hour to York from Kings Cross. Certainly, buying tickets once you get to KC for the next train is probably the safest insurance against a flight cancellation or delays or backups at Passport Control, or line delays on the Tube. This is assuming you aren't arriving on a bank holiday or school vacations. However you will pay the full, non discounted price., which is substantial for 4 adults. Buying advance tickets would provide a serious discount, but they are only good for that specific train. If you miss it for any reason, then you need to by a new ticket. Some people hedge their bets by giving themselves a couple of hours cushion factor, and buy their advance ticket for 4-5 hours after their Heathrow ETA. If everything goes perfectly, you could expect to get through Heathrow in an hour, then another hour for the tube to Kings Cross. If you find yourself with lots of time before the train leaves, you can have a nice long lunch and perhaps a walk.

I have a similar question (which may benefit the OP), and I'm hoping locals can help. On the LNER website, there are some dates greyed out in April (and May) which the website says is due to potential engineering works. Personally, I need to get to York on May 1st, which is the day before an extended bank holiday. Is there likely to be at least one train from Kings Cross to York despite works or the holiday? Or is there an alternative way to York (I don't care to drive).

(OP - April 25 and 26th are part of this, too. Not sure if those are your dates to travel!)

Just wanted to say that we faced the same situation but going to Canterbury not York. We did buy tickets ahead of time (there were three of us). We padded the time well and even after a leisurely lunch ended up a bit bored waiting. But that was better than stress!

Tickets in the UK are much cheaper in advance so for us it was worth it,

As regards the trains to York on 25/26 April engineering work dates are only firmed up 3 months before the date, so no-one (even LNER or Network Rail) currently knows for certain what is happening.
Thus we simply don't know what trains will be running.
For all we know the south end of the line (beyond Peterborough) may have a full line closure.
When that happens passengers are diverted to leave on East Midlands Trains or Thameslink from St Pancras to Bedford, then buses from Bedford to Peterborough or Huntingdon for trains north.
When that happens, then personally I would look at staying on East Midlands Trains to Sheffield, change for York; or even Avanti to Manchester, then Trans Pennine for York.
The former because it keeps me on trains and is much the same journey time as the bus/train combo, the latter because it will only be under an hour slower than the bus/train combo or via Sheffield, but it also keeps me off what will be very busy Sheffield bound trains.

Also Thameslink may be slower than EMT to Bedford but they are longer trains thus more capacity, and stop at other stations in Central London (London Bridge, Blackfriars, City Thameslink, Faringdon) so if taking the Bedford/Peterborough route I would probably prefer those trains from central London.

Great, thank you isn31c!