Can I start looking?

Good advice, Britwife. I would start by going through agencies anyway, as I don't have enough knowledge about working in the UK yet to feel comfortable applying to an employer directly.

I'd go and talk to an agency and get a feel for what the market is like. I wouldn't send your CV in because it's a perfect waste of time and paper. It's like Liz and Stacey said, they'll just bin it.

But do phone up an agency and get to know someone. They'll be a good contact when you are able and willing to work.

Just be aware that the Home Office may very well view this as you looking for work and it may affect your visa. Just a heads up.

Actually, I just got an email from someone at a recruitment agency requesting my CV. After I'd clearly told them that I was ineligible to work until after I get married in June. When I reply, I am going to very clearly restate my circumstances, just to make sure nobody gets in trouble.

I have to admit I was quite surprised. I was expecting a "Write back to us when you're allowed to work."

Maybe they just made a  mistake.

edit: I had my CV up on a few sites and was about to make it unsearchable, based on responses in this thread. Didn't want to get in trouble with immigration or anything. But too late, I've already been discovered.