I believe it will be worth reviewing what is Proof and what it is not. The governing legislation is "The Gun Barrel Proof Act of 1868". The goal of which was stated as protecting members of the General Public from the sale of guns that were inherently unsafe. In scope it covers both Domestic and Export Sales. It is updated periodically from time to time as conditions within the market change. In general terms it evaluates the ability of the submitted piece, Barrels, Action Body and Bolting system to withstand a single event overload/Per Barrel without damage or deformation.
I write this in the context of 1954 era proof, subsequent CIP edicts require metric markings and a double test per barrel. I've chosen 54 era for its simplicity and Imperial / inch units. As an aside the Rules of proof that apply to a gun are those in effect at the time of manufacture or most recent re-proof.
Of note here is it applies to the strength and integrity of the barrels ,action and bolting/(jointing). Nothing to do with forend iron tumblers,sears, safety, etc. Thus there is no such thing as a safety check other than barrel strength and so no "Safety Certificate " is ever issued, at least I've never seen or heard of one within the British gun trade.
For our working example Lets assume a Birmingham gun of 54 -89 manufacture.
A. barrels are viewed externally and internally for acceptable finish and lack of blemishes
B. A ground plug, approx 1&1/2" long of .729" is inserted to a depth of 9". It goes in!!
C. A ground plug, " " " .739" " " " " . It does NOT go in
This gun would be marked .729" NOTE it has not been measured at 729, it could conceivably be .738" but would be out of proof at .739" as proof is good for up to .010" above the Nominal dia marking.
A similar system of plugs measure chamber length in our example .800" and .810" . 2&1/2" to 2.550 depth would get a 12C (chamber) 2&1/2" length

Testing . Mean Service Pressure is 3 Tons PSI Lead crusher . Proof load would be 60-80% greater.vis 4.8-5.4 tons. The tester detonates by striking the firing pin or tumbler with a rod and hammer through a small hole in the rear of the test chamber.

Gun is the removed, inspected and marked.

So in answering Eds questions all we need to do is answer a basic binary question. IS THE GUN IN PROOF........YES/NO.?????

ED Goods Questions..
1. Do guns bound for export require Re Proof? NO guns offered for sale Need to be in Proof. Guns that are Out Of Proof require re-proofing.
1A.Shipped without Safety Certificate. As far as I know no such thing
2 Certainly don't cut antiques.? In proof NO/Out of Proof needs re proof prior to sale.
3. What constitutes an Antique in UK. I believe it depends on commercial availability of Ammunition. (Lagopus or Salopian could better qualify)
4.Do they use an arbitary date as US (12/31/)1898 ? NO see ans 3
5.Wall thickness part of Re-Proof? Wasn't part of original proof so No (other than if too thin may bulge or burst under re-proof)
6.What are the specs??? See Ans 5

I hope this helps, I, suffering fatigue and typers cramp
All the best,