Yea, you would start a new deductible, BUUUUUUUT I’ll give you a piece of advice:

When your new plan starts, call UHC and ask for a PRIOR CARRIER DEDUCTIBLE form. With a UHC to UHC plan, you should be able to roll over PCD.

It doesn’t always happen, but most of the time, they will allow it.

I struggle to believe that there is only 1 specialist in network and if there is and they have no availability until February, have your friend call their insurance to ask about an In-Network Exception, sometimes called a GAP Exception. Insurance companies will grant these if there are no available in-network providers within a certain distance with any reasonable availability.