I've thought about the tyres, and think that the best strategy is to wait until my current tyres wear out, then look at my best option for an upgrade.
I don't think it's worth replacing the rims until my current rims wear out. This is not meant to be my last ever bike, but something that I can use and learn more technical skills on. E.g. I've only recently started replacing my own spokes. If I could learn how to true wheels, then a future skill step might be to replace the current rims with thinner rims when the current ones wear out. Would that be an realistic aim?