I am not an expert on what has been changed but the essence of it is that the UK has been signed up to CIP for decades which means that we can accept guns proofed in the participating countries and they can accept our's.
However, the UK had been quietly ploughing its own little furrow and taking a pragmatic approach to the rules when applied to classic British game guns.
About a couple of years back there was some kind of shake up at the two proof houses and guns started failing proof where very few had done so before. Even the top gunmakers had problems with new guns and there was much banging of tables and frustrated phone calls.
The general feeling was that it was down to a change in proof loads but nothing was ever proven and with changes in choking (many of the unexpected fails where caused by bulging at the choke forcing cones), the situation got a bit better.
More recently, the CIP authorities took a hard look at what the UK proof houses were doing and insisted that they adopt a new set of gauges rather than using ones that date back to when God was a boy. These are super accurate and a whole new tranche of guns started to fail from over- or under-size chambers.
And yes, there has been a change in procedure although as far as I know it just means that now each gun barrel must be fired twice with the proof load, not just once. There is no double discharge test that I know of.
I have also experienced guns failed for mysterious and often invisible flaws!
I have heard it said that one proof house is staffed largely by ex-army and they apply the rules vigorously and to the letter. The other proof house is staffed largely by ex-guntrade people who take a sensible and pragmatic approach to the testing of old British guns as far as possible.
I couldn't possible comment on the veracity of this statement.
Sufficient to say that I know one barrel and action specialist who uses Birmingham Proof House exclusively unless ordered to do otherwise by their customer.