It used to be possible to deposit a third-party check/cheque here years ago by simply endorsing it over to somebody else, but it's become more and more restrictive over the years. The vertical or diagonal "crossing" with "A/C payee only" has already been mentioned. At one time these restrictions were added individually by the drawer if he felt it necessary; now they're pre-printed.
If you look at the payee line, you'll see that most, if not all, U.K. checks now also say "Pay _______ only" instead of "Pay _______ or Order" / "Pay to the order of ________."
The last time I asked my bank if they would issue me with uncrossed checks (and that was several years ago) I was told I could only get a book with about a dozen at a time, and that I would have to make a special request every time I wanted such a book.
You could ask, but it may well be that the banks won't even do that now.