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Changing wheel size will have implications on your brakes though (are they canti's?) Also you don't want a smaller wheel on the back than the front because no.
I suspect you can quite easily upgrade to more gears. I see downtube shifters, are they friction or indexed? If the former then it should work straight away with a new cassette and rear derailleur. Otherwise it might be worth updating the shifters to indexed 8 speed or higher.
If you do go for a new rear hub and need a new drum, I have an arai drum brake sitting in the cupboard that would like a new home.
Guys,
Thanks for all your help. I think a new wheel is out of the question because the current wheel size is 32-630 (27 * 1 1/4). - Is now the time to change this?
In terms of my preference for a screw-on 6-speed freewheel - I've had very little (no) experience with gears and am aware that nearly every part on the bike is bespoke, so thought the best options would be to try and keep as much of the wheel constant as possible. As has been alluded to upthread, this will likely mean slowly changing every part of the wheel in succession as it fails...
So... my next steps will be to determine whether my current drum brake is screw-on and to buy a new hub. Given the potential for a catastrophic failure and my wish to do some light touring with my significant other on this bike, I think it makes sense to change the hub.
@mdcc_tester - Thanks for your kind offer nonetheless.