London to York

I arranged for Blackberry Cars to pick us up at Heathrow after a 9:00 a.m. arrival. They drove us to the Kings Cross Station where we had a reservation for the 1:00 p.m. train to York. We had plenty of time to make the train, and were even able to grab a sandwich before boarding.

We just went to the train station and bought tickets the day we went. Plan for any eventuality. When we went, we were heading back to the York train station to return to London and noticed all the lights were off in that part of town. There was a power outage for three hours.

The train to York does not leave from Paddington. It leaves from Kings Cross.

Instead of taking the Heathrow Express, you can take the tube direct from Heathrow to Kings Cross on the Piccadilly line with no changes and for less money.

You can buy an all day ticket ahead off time and avoid the hassle of getting it at the station. But the price will be the same.

You can buy an advanced ticket but it is only good for the exact train you purchase. If you miss it, you lose the ticket and have to buy a new one.

Customs will just be a walk through. It's immigration/passport control that could take awhile. There is a difference between the two but many Americans combine them into one. Unfortunately, when you say "customs" to a European they will think you are only talking about customs.

Customs is for your stuff. Immigration/Passport control is for you.

You are far better off just buying a day single when you arrive.
Yes it'll be more expensive, but there is less of a difference than many people think between advance fares and walk up fares especially on that route.
It is far less stress. You are not in control of your airline changing the flight time and/or incurring delays. That is even before you reach the UK border.
Buying an advance fare to anywhere on the day of arrival off a long haul flight is always a gamble, with a good chance that it will be a false economy.

By the way, have you done the math on your whole planned train travel, and established whether it is best to buy a two together railcard, not senior railcards, for £30 to get 34% off all off peak train tickets.
2T is one railcard, senior would be one each, so twice the cost.
Just on the London to York trip it will almost certainly pay for itself.

Allow an hour to get clear of Heathrow, and an hour to take the Piccadilly line to King's Cross St. Pancras. Maybe allow a little extra - I have done this run in both directions a number of times and it's never been more than two hours. I would be more than comfortable booking a 1230 train out of King's Cross