If you’re not replacing the tires with it you should be able to have it done at any tire shop, or even at home by yourself. Deflate the tire and break the bead loose, this will allow you to pull the old tube out around the tire/wheel bead. Then insert the new one the same way. Be careful not to get any folds in the rubber when done or it can eventually leak or tear along them. a partial inflation and bouncing the tire while rotating seems to be they way they used to deal with that.
Also worth checking all the spokes are still tight, some people run some cloth tape around the rim to help protect the tube from any rough edges from where the spokes come through the rim…. the real issue isnt fitting as Mike explains.. finding a shop that can ballance wires properly is always the issue.
Are you replacing the tubes due to old age or leaks? If leaking make sure there are no manufacturing labels stuck on the inside of the tires-they will wear a hole in the tube eventually.
Don’t ask me how I know……Eric
I lucked out, asked the folks in the local MG club where they got theirs done, Gran Turismo East not too far from me. These folks could not be any nicer or do a better job, even said they can do the front end alignment after I re-build it. Actually knew what I was talking about as far as wire wheels.