I'm a brand new member here but long time trustee of small charities.
As part of switching from being a unincorporated organisation of a CIO, and registering as a full charity, we are considering tweaking our name.
Currently we are 'Aniridia Network UK' - Aniridia being the rare genetic eye condition we are concerned with.
The trouble is that we've taken on members in the Republic of Ireland too. Plus there's the potential for Scotland to leave the UK at some point. We'll presumably want to register with the different commissions in each part of the UK. 
There are similar organisations abroad so I don't anticipate us covering other countries any time soon. By the same token, having 'UK; in our name distinguishes us from them and is our established brand.
I wondered if any one had any thoughts, advice or research into the pros and cons of including 'UK' in their charity name, and if not what the alternatives might be.  

Thanks for any insight you can give