Now that the workshop has come up to a manageable temperature, I've been back to check actual dims, both rims are within a gnat's toozle of 22 5/8" o/d. The ancient rim is 1 13/16" wide, the new one is 2" wide, so apologies for wildly inaccurate guestimations! I also found a newer rim that I got from the VMCC supposedly for 26 x 2, that also measures 22 5/8" x 1 13/16", but the welded joint leaves a lot to be desired.
It looks like I may end up ditching the old rims, reworking/improving the VMCC rim (so I can get it nickel plated for the stirrup brake) using the new slightly wider rim on the rear, and as you say, sticking with 2" tyres (my other 1912 little Enfield twin has 2" tyres, but they are the old delivery push bike tyres, on wired on rims). No doubt fitted in the days when BE sizes were even scarcer than today.....
2" tyres at 40+ psi are no joke - I know from riding the other Enfield! But great fun until the pain barrier is reached.
Still seems odd that same o/d rims fit different section tyres........