Yes I regularly had half a ticket as a child, They were always single tickets. I never had half a return ticket. I don't remember them cut at an angle, mine were always cut straight across. Presumably as the hald that was kept could not go back in the rack, these were cancelled and kept to account for the shortage in fare money collected rather than issued to another child.
If dedicated child tickets were not held at a station, the rule was that
singles would be cut straight across the middle, and
returns would be cut diagonally. Presumably the idea was that the whole half single would be surrendered at the end of the journey, while only the tiny sliver across the halfway mark would need to be given up on completion of the outward journey - though I have no idea if it ever was in practice!
This is why, on traditional Edmonson card single tickets, the journey was repeated in small type on the left and right hand sides of the ticket. If you sold a half ticket, the other half
could also be sold - and you always hoped it would be, before the end of the accounting period, as it was much less of a faff than cancelling and accounting for odd halves, especially if odd half-pence were involved too!
Around the late 70s, BR redesigned their Edmonsons so that the journey was only printed once, and left-justified on the ticket. I was dismayed because it meant you could no longer cut them in half for a child ticket - and the alternative was having to write out a paper ticket. Fortunately, I found out that you could still get tickets printed in the old format if you requested it on the order form. As I much preferred the look of the old style tickets, I made sure to order them, even if it wasn't strictly necessary (where we had actual child tickets for a journey, or where the child fare wasn't half the adult). Years later, I happened to meet someone who had worked in the ticket supply section at Crewe. He said, "Ah, so you're the **** who kept ordering old-style tickets!"
