Would have thought it would depend more on how your current / recent employer is set up than on TFL. Some places are too small to have an HR department, so anything would have to go to manager / director / owner.
I've worked in large organisations where it's been policy that any reference request has to go through HR (even if it is addressed to an individual manager) and they may prefer it to go through a group mail inbox (as in
[email protected]) rather than to a specific named HR person.
An increasing number of employers won't give a detailed reference for people, they will just confirm dates of employment and job title, and not a lot else. (It is an urban myth that "it's illegal to give a bad reference" - but a reference that's false or malicious can be the subject of legal action. Many employers choose not to take any risk here.)
I've also got as far as the references / background check stage with one employer where they were happy to accept something I downloaded from the inland revenue bit of government website which showed I'd been employed for the last X years, without even bothering my former employer (not that I had a particular problem seeking this, but it was quicker all round that way.)