Thanks to the answers to this recent question of mine, I can now check many products including some breads but not all; for example, not the bakery in Tesco.
I was reading the labels and signs carefully in the Tesco bakery but the answer was not clear. I could not see any obvious non-vegan ingredient but nor any statement that some products were vegan. One possible explanation is that there was a generic warning of cross-contamination. So, it might be that some items had no non-vegan ingredients deliberately added but some contamination from other products was possible. This is a situation that I would accept.
I have not tried talking to the staff yet (I did not have time on that visit).
Does anyone know the status of bread in UK supermarket bakeries?