Trains are owned by Rolling Stock Companies (ROSCOs) the various Train Operating Companies (TOCs) then lease the trains from the ROSCOs. The exact terms of the lease can vary, some might literally be just for the train itself, others can include the ROSCO handling the maintenance and simply ensuring the trains are ready for service each morning.
Back in the early years of privatisation new rolling stock was primarily the responsibility of the individual TOCs. They were expected to procure, with government backing, any new trains they felt would be needed. So for example, Virgin took the lead in procuring the 390s for the WCML. Around the same time as the SRA was dissolved (2006) the government decided that the DfT should be responsible for rolling stock procurement. This has most clearly been seen in the continuing IEP debacle which is an entirely government run procurement program with TOCs just being told what trains they'll be getting. The current government have stated that they want to move rolling stock back to the TOCs, but whether or not they'll actually follow through on that remains to be seen.
So in general terms, trains are owned by ROSCOs. They are then leased by TOCs on a variety of different contracts, some might include the cost of maintenance, some might just be for train with the TOC covering maintenance. Procurement, at the moment is handled by the government through the DfT. TOCs can lobby the DfT for new stock, but they only get it if the DfT think the numbers add up.
