My advice on this is as 0844 815 is an Opal Telecom (part of Talk Talk) prefix, then it could well be an NEG system.
If a new number isn't on the internet anywhere, then it's not going to be found. In general, I believe that these systems provide a block of 01/02 numbers for the surgery and one of them is the main one that the 0844 number directs to. Other numbers are likely to be direct to extensions for staff. Find one and you might be able to get to the one under the 0844 number.
A more detailed explanation is on
this request for another dentist. Have a look round the surgery next time you're there; see if you can find any 0115 numbers on the wall or on someone's desk and make a note of them.
The old number that redirects to the 0844 number is 0115 939 8063.
Do they have a website or email address?