Do flaxseeds (linseeds) contain vitamin B12? The short answer: (quite probably) they don't.
I came across a publication from the 1990s (on PubMed) titled "Process-induced compositional changes of flaxseed" (Wanasundara & Shahidi 1998). The paper includes a table that lists the mineral and vitamin content of flaxseeds grown in Canada, and vitamin B12 is also listed, with a content of 0.5 mg (!) of vitamin B12 per 100 g of flaxseeds.
0.5 mg (milligrams) are 500 µg (micrograms).
Even though eating 100 g of flaxseeds would be a lot, eating several tablesspoons of ground flaxseeds in baked or cooked goods is possible, and 50 µg in 10 g of flaxseeds would be a very high amount. See their Table 8 (Wanasundara & Shahidi 1998) below:journal that is not PubMed-indexed) by Bhatty & Cherdkiatgumchai (1990), also based in Canada, titled "Compositional analysis of laboratory‐prepared and commercial samples of linseed meal and of hull isolated from flax". Bhatty & Cherdkiatgumchai did measure the B12 content of flax. But I would guess that (1) they mixed up the unit and wrote "mg" instead of "µg" and that (2) the B12 content they found is an artifact that may have arisen because of issues with their methodology (they give three general references for all of the analyses - which I did not check) and/or it may have been caused by contamination with B12 from other sources.
I have not been able to find any other papers that include information on vitamin B12 in flaxseeds. But until anyone else publishes any data that contradicts the assumption that there is absolutely no B12 in flaxseeds, we are probably correct if we assume just that.
Moral of the story: Peer review could do better ... and better triple-check everything you read.