I am afraid that I don't have a proven answer to these questions.
Here is what I think:
- Purdey marks tube numbers, one per tube on the Forend loop.
- Kilby marks tube assemblies, one # per assembly (marked on each tube)
- Bernard made and marked tube assemblies, one # per assembly.

I guess the practice depends on what was sourced outside, tubes or assembled tubes.

In the case of Purdey, I have no doubt that Purdey did the marking. The forend loop is made by Purdey, and is independent from the tubes.
The process is no different from marking Patent Use Numbers (PUNs). There had to be a ledger at the maker. The process existed.

In the case of Bernard, he was for sure providing assembled tubes, so kept track of the numbers himself.

Best regards,
WC-