As with the other elements of funding in the government's undergraduate student finance package (i.e. grants for dependants, Disabled Student Allowance, and the pre-2016 Maintenance Grants), the payments are made by the Student Loans Company (SLC).
Their head office is in Glasgow. Information about the Company can be found on their website - www.slc.co.uk.
At the moment, applications can be made either online through the government's Student Finance website - www.direct.gov.uk/student-finance - or by a paper application form (PN1 for new undergraduate students, and PR1 for continuing students). For students in England, a contact centre authority in Darlington called Student Finance England will assess your application.
The Student Finance contact centre will assess your application to see how much you are entitled to receive - and they will then tell the Student Loans Company how much to pay you. Any payments due to you will be paid into your bank account on a termly basis. Tuition Fee Loans are paid directly to the University.
Contact details for Student Finance England can be found here