They're in the highway code, but the word "must" isn't used in the online version or my paper copy from 1999, which states:
155. Well before you turn right you should
Use your mirrors...
Give a right turn signal
A cycling-specific rule is 52 in my copy (67 in a more recent edition):
52. You should
- look all around before moving away from the kerb, turning or manoeuvring, to make sure it is safe to do so. Give a signal to show other road users what you intend to do.
We could argue about whether the should applies to giving a signal -- it certainly would if it was in the first sentence of that bullet point, or a bullet point in its own right, but we dont need to: in the context of the highway code, context must means a legal requirement, while should means if you don't and there's an accident you're likely to get at least some blame.
So no, you're not obliged to.
In fact, in one of these cases (moving away from the kerb) I tend not signal -- because I wait for a big gap, then as the vehicle at the front of the gap passes me, I accelerate hard, with both hands on the bars for control and changing up through the gears. Signalling after it's passed wastes some of the gap, and theres often no room to signal while it's passing (signalling before it passes would only confuse its driver).
However if there might possibly be traffic about, it's a really good idea, especially signalling right, as that means you're about to move into the path of faster vehicles. If you're right handed this may take some practice. So practice somewhere quiet; you'll soon get the hang of it. Try riding left handed with your right hand near the bars, for example. Until you can confidently signal right there may be cases where the only way to go right safely is to stop and cross like a pedestrian.
Never ever assume that drivers react to you looking round. Hardly any will notice, and they may take their eyes of the road to see what you're looking at. Even with helmet lights that clearly indicate me turning my head (red rear, amber sides) I've often been overtaken while getting ready for a right signal.
Drivers don't understand cyclists' speed. You're going so slowly from their point of view that you'd have to be down to walking pace for them to notice. Moving out into the lane can help. But you need a good couple of hundred metres gap and visibility even at 30 mph -- I'd signal then too.
A comment on my answer to a related question confirmed that a slowing down signal is very rare. So don't worry about that one.