" I will pick up the photos at half four. "

I hear it from only British. Not from Aussies, Kiwis or Americans.

Do you normally omit the 'past' in 'half past'? I mean is that British style?

Thanks in advance.
I don't know any Americans who would use the British "half four".
We'd say "four thirty" or "half past four".
Excluding the 'past' is more informal.In the UK you'll hear both versions in speech, but we only write with the 'past' included.
Students: We have free audio pronunciation exercises.
I had British neighbors for a few years when I was in Germany, and we used to have regular misunderstandings about time.
In German the word for half is halb and the word for four is vier. However, if you say "halb vier", it means 3:30. This resulted in endless mix-ups with my British neighbors over what time we had agreed to do something.
Students: Are you brave enough to let our tutors analyse your pronunciation?