I have always used CR1632 in the key fobs.
UPDATE: Not sure why I wrote this at the time. Others have reported that the CR1632 (which is 3.2mm high) does not fit in the NC key fob. I’ve just re-checked my key fob, and I have indeed used a CR1620 (which is 2.0mm high) in it.
It’s a 30 second job - no need to use Timpson or anybody else.
Are you certain that your CR1620s are genuine Panasonic? A large percentage of the “branded” batteries sold on eBay, Amazon, AliExpress, etc are fake.