If it's in the south of England, it's hardly been cold. The fact the month's called January doesn't make it less suitable - it's often colder in October and plenty of people caravan then who wouldn't normally consider doing so in January.
Is it entirely normal for a 12 day old to be away from its mum with granny for a weekend? No, but so what? I would assume mum was struggling really, really badly and needed a complete break (not just baby in the next room with another adult). Everyone doesn't bond in the same way. The world does not end if you are not the only person your baby trusts.
And yes, if a friend told me this I'd assume it was a sideways request for support.
The whole "it doesn't have central heating or double glazing" thing is very odd. I'm trying to imagine the conversation.
Friend: Baby spent the weekend with mum.
OP: Oh really, were you okay with that?
Friend: Sure, I know the caravan is really old and has no double glazing or heating but I knew baby would be fine...
Or is it far more likely to have been "sure, the caravan's perfectly warm and cozy"?
You can't possibly tell from a photo whether a caravan has double glazing. We had a caravan where some windows were and some weren't and from the outside they all looked exactly the same. Similarly for central heating - it's often hot air blowers at floor level, they look exactly like ventilation grates.