More likely there would be no timetabling or fare integration at all; train operators would run their own ticket offices which would only issue tickets for their own lines, except where they have decided to co-operate. You would have to book at several different places for a through journey. At major termini and interchanges, you would have to know which company's ticket office to go to in order to buy a ticket for your destination.
This is what was originally proposed for privatisation. It was envisaged that there would be several companies running trains between, say, Reading and London, all of whom would be in competition with each other for custom.
If it had been introduced like this, it would have culminated in mergers and buy outs, with the final effect of being a few dominant operators across the UK forming a cartel!