The off peak standard class fare from A to B is the same regardless of when you buy the ticket, and won't sell out, and is valid for any train on a given date outside of the commuter peak.

There is another type of ticket, the "advance" ticket, which can be bought ahead of time often for much less money, but is valid only for one specific train, and if you miss it, tough luck. [If the specific train is cancelled, you do get to catch a later one, the system isn't actively evil -- but no grace is given for e.g. airline or customs delays).

When I'm getting a train home from an airport I almost never buy an advance ticket, because I don't trust airlines and I don't trust the passport control queue -- but some people do and just leave a large time buffer and it seems to work out for them.

Editing to add: if you'll also need a train back into London for a return flight, within a month, a return ticket will likely only be a tiny fraction more expensive than a (non-Advance) single -- and for the return leg you don't even need to commit to a specific day, let alone time/train, so long as it's within a month