There's discussion on this subject here. (It's from a landlord's point of view but still relevant anyway, I feel):
https://www.property118.com/taking-rent-tenant-overlap/
The answer provided by Kate Mellor, on that thread, seems to sum up the legal position quite well. i.e. by accepting the new tenancy, the landlord has, in effect, agreed to release you from your own tenancy early.
Before considering any formal route to get a refund, simply write to the landlord (and/or letting agent), pointing out that the landlord ended your tenancy early and that you therefore believe that you are due a refund of your rent for the relevant period.