There are lots of products on the market which address this problem. You can get lovely fleece, welly liners. I'm very fond of my long Cath Kidston welly socks which have to be the mother of all warm long woolly socks. They stay up, have nice designs and are the ticket for cold weather activities under any pair of trousers.
However my feet remain cold. So I thought it would be fun to see if I could make myself a pair of insulating soles from felt. This has been inspired very much by Yuli Somme's Felt Feet. The problem is that the feet are too tight to get my feet back into. Thus I went for insoles instead.
The process is pretty basic. I went for the same approach as I had used to make portable felt seats. So have look at this post to see the technique I used. I was actually a little bit disappointed in that I rather liked the outcome below and it would have made a lovely seat.
However, needs must! Luckily I had an insole of the correct size for my welly boots. So I carefully cut around the outline to create woolly insoles. I had enough felt for another pair of wellies too!
Once the welly warmers were made, I slipped them into my wellies and took my dog for a walk. I was wearing thick socks too. Initially I was worried that the bobbly felt surface would cause me to get blisters but this didn't happen.