IME, no it won't trigger NHS treatment.
I also have Hashimoto's. 2 years ago I went to the GP complaining about being permanently worn down and exhausted and they tested B12 and folate in addition to TSH and FBC. Both were below range.
B12 was 161 (ref 180-1000), folate was 3 (ref 4+)
They ordered repeat tests straightaway to check the results and mentioned the possibility of B12 injections, but at the repeat test 2 weeks later the B12 measured 208.
I was put on 3 months high dose folic acid (5mg daily) which sorted the folate and brought it up to 11.4.
No treatment offered for the B12 as it was just into range, just a retest in 6 months. I researched and decided to self treat. I take Solgar 1000ug sublingual B12 nuggets most days.
6 months later B12 was 617 so the pills clearly helped.
I am now on the system to get an annual B12 and folate check. This year's was 409 so it has dropped again, but I have been less reliably taking the pills so I think they are important.
The Pernicious Anaemia society has an active forum which may be helpful?
I was tested for PA due to the low B12 and that test was negative. You could ask for that on the basis of both your low B12 and your existing autoimmune diseases - PA is autoimmune as well. The test needed is the one where they check for "Intrinsic Factor". However, like coeliacs, it's fairly common to have PA but not get a positive test result, so it still doesn't completely rule it out. If you had PA you would need B12 injections regularly for life and would get those offered.
If you don't have PA, then personally if I was you I would look to supplement myself. B12 is best absorbed direct into the bloodstream, either by the injections the GP's prescribe or through sublingual nuggets that you dissolve under the tongue. If your levels are low, and esp given you have coeliac disease, it's likely that you are not adequately absorbing dietary B12 so this should really help.
Have you also had your folate and your iron levels checked? Because deficiencies in either of these can cause and/or worsen B12 deficiencies, the three minerals work together. I am hazy on the details now, it;s a couple of years since I read up on it, but I do remember that it's important to consider all three results together not one in isolation.
Good luck.