SOGA no longer applies, it's now the CRA.
They do not have to supply you with an invoice, nor a letter stating the work that was carried out. Of course, it's good practice to, but they are under no obligation.
30 day warranty means zip, unless you've bought a Cat B car for £50 then the CRA covers you. Tbh at this point I'd be booking an appointment with the dealer principal (if a small-ish garage) or the sales manager (if bigger), and show them the footage. Ask them if they think that's acceptable (remembering that it's not their employees that did the work, so they can't exactly fire them!) and point out what still needs doing. Quite simply, if you don't get the response you want then the next step is to release that footage to the media, as they'll lap it up.
Did you at least point out the issues within the 30 days?