"he has a cold" = what you expect.
"he's a cold" = "he has a cold" for some British speakers = the same problem as for gonna wanna shoulda etc that many learners take many years to understand and sound ridiculous when actually trying to replicate the sound; discussed frequently on this discussion forum.
"he's a cold" = also it means he is a cold (whatever) "he's a cold son of a female dog"
That's the only over way I have heard it but it looks doubtful this is the case for the situation being discussed, so I assume it is the British accent for many British speakers causing the problem. Even though I am a Londoner often having to endure jokes and bad London speaking imitations I personally always say "he has a cold" taking care to separate the "he" and "has - a".
This has nothing to do with the stereotypical cockney dropping of "h's" that occurs only on some words. The two most common ones being "house" and "hill" "'ouse' and "'ill".
And @Liam Patrick also has a valid explanation of what might be happening for a regional British dialect.
Also I have heard @Mathew Beardwell''s example both in London and in more rural areas