The UK had/has far too many companies following the 1990s privatization, but any company can sell a ticket on any route -- online, at a ticket office, or using a machine.

I don't know the rules, but at least the company with the cheaper/slower trains from London to Birmingham shows their faster/dearer competitor when searching, and vice-versa, although you do need to search "London Terminals" to "Birmingham Stations" rather than specific stations. (The trains use different stations.)