Britrail Pass

The short answer is that there are NO daytime trains in the UK which require reservations. Although they will try very hard to tell you otherwise that includes the open access operator LUMO.
The night trains to Scotland and Cornwall are the only compulsory reservation ones.

You can always just walk on to any train. That includes a York to London train.

However, there are cases, such as York to London, where there may be no seats left-- especially on a train which started from Newcastle, Edinburgh or beyond. Every 2 hours, an LNER service starts at York, so that will certainly have seats.

But even an LNER train always has part of one car with unreserved seats (Car C for standard class). So booking is strongly advised.

The golden rule is never pay ACP to make reservations for you. Reservations can always be made for free at any staffed station (for any train in the country). So you could reserve at Edinburgh for a London to Cornwall train for instance.

For LNER you can also reserve on line direct with them up to 10 minutes before departure- https://www.lner.co.uk/travel-information/make-a-reservation/

For any other reserveable train in the UK you can make your reservations on line for free with either GWR or TPE.
In both cases you need to create an account with them,
https://auth.gwr.com/gwr/registrations?client_id=640e58c3-8156-4149-990d-65e96b973da8&redirect_uri=https%3A%2F%2Fwww.gwr.com%2Fbuy-train-tickets%2Flogin&response_type=code&scope=openid&code_challenge=36PGqU_nkGF1fvwCF5RlNsqpiW8QKDg1exlBkh8q7jY&code_challenge_method=S256
https://ticket.tpexpress.co.uk/widget-auth/register?redirect=tp