Am sorry to read about your Mum. With a FT4 that low - it does suggest her T3 will also be VERY low too. It is possible that the steroids will affect her thyroid readings. You have not posted the ranges - so am guessing based on other posts
Don't let them stop her injections - she will need them regularly. The NHS protocol is every 3 months - but if symptoms return quickly - she MUST have them more often. The new Guidelines state that GP's should look at clinical symptoms and not the blood test.
There are sprays for B12 - that can be sprayed under the tongue for better absorption. She could use it every day to help out with her levels. In between injections. It is my non medical opinion that all older people are low in B12 as the stomach acid lowers as we age so it is difficult to grab the B12 from food for the onward journey from the stomach. She will also need to take a GOOD B Complex to keep all the B's in balance as B12 works with Folic Acid in the body.
Has she had her VitD tested ? Ferritin - Folate too ? They all need to be at a good level for your Mum to feel well.
Please take time to look at the above website - as there other tests linked to B12. Also the videos under the heading Films are very interesting and illustrate the importance of B12.
Hope all goes well....