Warm, compact, cheap: pick two.
You don’t say what time of year you intend to use the bag, a big difference in the UK between summer and winter, for the price you mention I’ll assume summer pushing in to spring and autumn but not the cooler months of those seasons.
For the best warmth to weight ratio go for a down bag, the downside (sic) is that for a decent quality bag you will pay more. Also for a given weight/warmth down is less bulky when packed than synthetic filling. A good quality down bag will last a lot longer than a synthetic bag, a poor bag can lose up to 20% of its warmth per year depending on use.
Another option is a quilt, not everyone gets on with them but they are lighter and a bit cheaper than an equivalently rated sleeping bag as they don’t have down on the section underneath you. You do need a reasonable sleeping mat to use with them though.
My current system is a 2 season down sleeping bag (rated to 5C) which is suitable for quite a chunk of the year as I’m a warm sleeper. I augment that with a similarly rated quilt during the cooler months, the system is good to around -5C at least (I haven’t had chance to test it in anything cooler yet) and is much more flexible than a -5C rated bag would be. It does cost though.
With a budget of £100 I’d look at second hand as I think you’ll struggle to get anything that isn’t massively compromised, one range that might be worth a look is Blacks – they do DofE kit but it can be quite bulky which you’ll notice when packing on a bike.