I agree, should be just fine.
Just think, probably 95% of buyers of a rifle such as your BSA would just load up any modern 303Brit. ammo they could get and go on their merry way.
Obviously, you are a more educated shooter (you are here on NE after all) so if you are really concerned, I would suggest that you could shoot with a .308" bullet and then measure the ID of your brass and come up with an optimal bullet diameter for which your chamber will allow.
You will probably find that the standard .311" will work out just fine.
Super nice rifle BTW.
Gungadoug pipped me on a Westley retailed Lee Speed just last week which Tinker advertised right here on NE.
Now... I bet Doug will just LOVE that rifle and it will never leave his collection although I have caught him in a weak moment before...

Once you get that Speed running, I bet it will become one of your favorite rifles.