The cheap "magic", one-size-fits-all mittens are as good as any for UK winters (use two pairs if it's particularly cold). You can buy them at primark or on the markets. I sew them onto a length of elastic and thread them through their coat sleeves but you can also buy fancy clips to clip them onto their sleeves. Because they are cheap, it doesn't matter quite as much if you lose one or they refuse to wear them as it would if you'd spent £££ on fancy ones from jojo maman bebe for example.
We got some gloves in H&M that came as two pairs, one fingerless and one full pair. Her hands are quite small so the fingerless ones cover quite well. They are Frozen branded, but they may do plain ones. It's a bit of a pain getting them on but we found mitten ones tended to have the thumb in the wrong place so she just had her whole hand in the mitten.