Now, I have no idea whether this would be allowed or not. Is there a
way for me to get something like "approval in principle" before
actually buying the house???

You can apply for planning permission before buying the property. You need
the permission of the current owner and bag of gold for the council. You
might even be able to get rapid outline approval before purchase and fill in
the details afterwards, although I don't know if you need to have a much
larger application to be allowed to do that.

As for extending over the garage, it will probably be simpler to bulldoze
the existing and rebuild that section. The underpinning required to the
garage to be allowed an additional storey would likely be more expensive
than a scrape and dig. You can also rebuild using the appropriate bricks to
match the house, cavity walls, insulation and fireproofing (if you're
keeping it as a garage).

It's a bit scary to think that I'll be takign the side of the house
off! How does this get done without the whole thing falling down?!

Anything's possible with the right structural engineer and enough cash.
It'll involve a lot of steelwork, though, which may intrude slightly into
the living space.

Christian.