I do it/did it.
My setup was a child's seat on the bike. I started with one that was in front of me (Yepp Mini is a good, comfortable brand) and then when she got older moved to a Yepp Maxi. Now that she's four I have an xtracycle that she rides on the back of.
In the winter (below 35F or so) I have a trailer that I attach. It does get kind of cold or maybe my kid is just a wimp. Anyway, the trailer with a blanket is always warm enough for her and she's never complained, even if it was -10F.
In the rain you can get child's rain ponchos that are made to fit over their bike seat (I think bobike makes one?). I myself also use a poncho in the rain. I just keep them in my pannier.
Oh, and if it is icy, you really have a couple of options. I have studded tires, but I doubt they would be useful many times of the year in England. So you could just walk instead. If its icy, I always have the kid in the trailer rather than bike seat because I really, really don't want her to fall off the bike. She could get hurt, obviously, and it would be traumatic. I've never actually fallen with her on the bike when it is icy, and like I said, I have studded tires, but its something to consider. I have fallen alone without the studded tires, and once my back tire was slipping, I was down before I knew what happened. The kid on the bike really changes the handling, so you need to be prepared for that.