The "pressure" seems to be from the teachers themselves who are the one stressing out the kids.
We ALL sat the 11+ when I was at school - I remember no pressure despite them being far more important that today's SATs - i.e. your secondary school was determined by your 11+. The teachers taught the stuff you needed to know as part of normal lessons, typical questions were worked through in the classroom beforehand, etc. I don't actually think we knew what it was when we sat it - I certainly don't have any strong memories of the day one way or another, I don't recall any stress or anxiety any more than any other test we did. Just a normal day at school.
I see the contrast in different teaching styles today with my son at secondary school. "Some" teachers really stress him out when tests approach - they hand out loads of homework, worksheets, revision papers, etc in some kind of emergency last minute scenario to cover what they didn't teach in lessons - a lot of his "revision" for these teachers is learning stuff that's not actually been taught - I don't call that "revision" at all!!! Other teachers barely mention it, but those are the ones who've been consistent over the term in giving regular homeworks, regular short progress tests, etc., so their "revision" is a lot quicker and simpler and is what I'd call "proper" revision, i.e. recap the work you've done and what you've already learned.