"I am wanting a new bicycle."
Translation:Tha mi ag iarraidh rothair ùr.
There is overlap, but if you said to me "tha mi ag iarraidh rothair ùr", I would infer that a) you already have a bicycle, and you want a replacement/ an additional one, b) you may or may not be happy for the rothair ùr to be the same model as the one you already have c) you want it to be bought straight from a shop/the manufacturer, etc., i.e. you don't want it to be secondhand.
If you said to me "tha mi ag iarraidh rothair nuadh" I would infer that a) you may or may not already have a bicycle, b) you want a bicycle that is the latest model with more functionality than bikes have had up until now, c) you may or may not care whether it is bought straight from the shop or secondhand from eBay.
So I guess I would use ùr for something new as in fresh, not used before, replacing something worn-out, etc., while nuadh is something new as in modern, up-to-date, replacing something by being qualitatively different from existing models.
Thanks for the explanation :-)
It sounds a little like the difference between French nouveau and neuf.
I can never remember which way round it is, but one would mean "the bicycle is new to me, but it might already be old and secondhand", and the other means "the bicycle is brand spanking new".
So ùr means something like new to me, but nuadh means something like brand new?