If I was on holiday and my child was born, would they be legally entitled to citizenship of that country?

No, like the UK, Germany does not have birthright citizenship, the child will only be a British citizen

In order to claim German citzienship, one or both parents must be German citizens

You ought to check if the NHS has a reciprocal agreement with the German authorities which will entitle your girlfriend to free medical care if she does give birth - make sure both of you have a European Health Insurance Card

Your child could only get German citizenship if (a) at least one of the parents is a German citizen or (b) at least one of the parents is a legal permanent resident of Germany.

There are no exceptions, Germany is not dumb enough to hand out citizenship to everyone born in its borders.

German citizenship is (generally) only if one of the parents are German, or is a permanent (legal) resident for Germany for three years.

If your gf was born in Britain and keeps her British citizenship, then her children are British. It doesn't matter where they are born. But the details can be complicated.

I think the only country that gives automatic citizenship to babies born while foreigners are in country is the USA. That's why so many illegals come here from south of our border.

it depends on the countries immifration rules, in the uk one parent has to be british for the child to be british, if the both parents are another nationality eg chinese then the child is chinese unless the parents have ilr

Yes he would be classed as a German citizen regardless of u and ur gf being british