If you've been there for less than a month, you can be dismissed without notice. If you've been there for more than a month, you are entitled to one week's notice. Your contract might give you more rights than that, but that's the legal minimum.

It probably isn't in your current employer's interests to sack you just because you plan on leaving, because it's not easy to find staff at the moment. Notice cuts both ways and they'll want time to find someone to replace you. With that said, you'll probably want to keep quiet about it until you're sure that you have another job to go to. Nobody wants to be the person who tells their boss where to stick it, only to come crawling back when their offer falls through. You might not get sacked if it's known that you want to leave, but you definitely won't be up for promotion.