You’ve got a good option that doesn’t require taking the Tube or a taxi/car all the way from Heathrow. But first, have you gotten a Two Together rail pass? It gives a substantial discount on train tickets throughout the UK, any times outside of morning and evening Rush Hour, and would work for your fare from Kings Cross Station to York, plus any other overground trains you might want to take this trip, or in the next year. This includes the Heathrow Express train, that gets you from Heathrow to London in 15 quick minutes (more on that in a moment).
The Two Together Pass is good for an entire year, and costs just £35 for two people. If you have any train plans besides getting to York, and if you haven’t already gotten your York tickets (or if you have tickets, but they’re refundable and you can repurchase cheaper tickets with the Pass), consider getting it. You can download the Pass onto your phone, and the Website is https://www.twotogether-railcard.co.uk/using-your-railcard/where-to-buy/
Now, about the Heathrow Express, which didn’t exist 26 years ago. It collects passengers at Heathrow terminals, and in 15 minutes arrives at Paddington Station in London. That doesn’t quite get you to Kings Cross Station, but a Black Cab taxi can get you the remaining stretch from Paddington to Kings Cross, a shorter and cheaper ride than if you’d come all the way from Heathrow by taxi. The Heathrow Express is cheaper the earlier you book, and cheaper if you opt for non-refundable tickets. Tickets a couple weeks out would normally cost £44 for two people, but are less than £30 total, using the Two Together Pass. Download the Heathrow Express (HEx) app, select the tickets you want (making sure that you pick Heathrow>Paddington, and not going the other direction), then on the next screen, “Adjust your ticket,” select the Railcards box, and click on one Two Together (or more, if there are more than two pairs in your group with passes). With the HEx and a Two Together, that one ride is already saving you £14 towards the £35 Pass price, in a quick, quiet, roomy ride into London. Finish it with a short taxi to Kings Cross, and then you’ve got a UK discount railcard for discounts on journeys for the next 12 months.