I don't think this is entirely correct. The various EU rail packages have required a degree of competition to be introduced into the rail sector. It also (and perhaps notoriously) requires the separation of infrastructure from operations. But the devil in the detail is that the latter is for accounting purposes only. In simple terms it is intended to ensure that, having created separate legal entities to manage infrastructure and operations and in order to provide a level playing field in terms of access to the network, a state owned operator does not subsidise one legal entity with the other. If anything the EU copied Sweden who were the first to separate infrastructure from operations in the late 1980s.
The separation of infrastructure from operations has been applied differently across the EU. Sweden, Netherlands and the UK opting for separate ownership with others opting for an infrastructure company and operating company owned by and overarching holding company. The holding companies in all cases accross the EU are state owned, although in some cases, the state owned holding companies are legally private limited companies with the state being sold shareholder.
The UK (for political reasons) interpreted the EU legislation in the most extreme and far reaching way by entirely destroying the state owned operator. This is extremely ironic because after sectorisation, BR was probably already heading in a direction that wouldn't have required major surgery to comply with various EU rail packages.
The competition the EU legislation effectively mandates applies quite differently depending on the types of rail services. Focusing specifically on passenger operations:
1) Public Service Obligation type services (i.e. those that are likely to require some kind of subsidy, typically regional and local services) are required to be put out to competitive tender. There is, however, nothing to stop a state owned company bidding for the contract and operating the service. In Germany, DB holds between half and two thirds of contracts, which are awarded by the Lander or by location transport associations (equivalent to English ITAs). Outside of Scotland, Wales, London and Merseyside, the UK is actually very unusual in the very limited role it actually gives to local authorities in respect of local and regional railways.
2) For long distance/commercial rail services the EU legislation (as I understand it at least) envisages a more open access model is to be used. This does not prevent there being a state owned operator. Indeed across the EU many state owned operators run long distance services on a commercial (as opposed to franchised/contracted basis) and face some competition in doing so from other operators, whether they be entirely privately owned operators or the state owned operators from other countries.
In addition to smashing BR to smithereens, the UK implemented a franchising model for all passenger services, with some limited open access at the margins.
If we'd have taken a more incrementalist and pragmatic approach to applying EU law then it isn't impossible to imaging BR having evolved in the following way:
1) Regional Railways and NSE putting out service groups to tender, possibly based on the same profit centres that in reality did become franchises, but RR and NSE continuing to exist for branding purposes, rolling stock ownership and acquisition, marketing, overall strategy, station management and potentially even infrastructure on some routes.
2) Intercity being established as a limited company, either wholly or partially owned by a BR holding company, operating on an entirely commercial basis;
3) Open access operators competing with Intercity on key long distance routes.
4) Infrastructure held by a separate entity owned by a BR holding company, either in its entirety or with some delegation of responsibility to RR or NSE where they are the sole operator on certain routes.
In answering the original question, under current circumstances or under the alternative history referred to above, I'd envisage open access operators competing with the incumbent/dominant operator on the East Coast, West Coast, Midland and Great Western Mainlines and potentially some core Cross Country routes as well.
