Here in Scotland we've had the charge for a while now. Nothing happened when it was introduced except some folks started taking their own bags and some folks paid 5p. The sky didn't fall in! 
I've been using my own bags for years and tbh, it's actually rather nice not to be questioned as to why I do that (Primark were very snippy) or, with M&S at one time, they wouldn't allow you to Collect by Car if you weren't using their bags.
The one issue I have is with meat and fish, as I do think for hygiene reasons that should be wrapped but what I do now is put it in my bag and then chuck the bag into the washing machine. Easy!
I thought I would miss bags to some extent for rubbish but even where I live in the depths of the countryside, we now have recycling of everything, including food waste, with the exception of glass so we don't have much proper rubbish. I don't mind taking glass to the recycling centre as I think it's fun to chuck it in and hear the noise it makes. 