Denturetablets are the answer - works beautifully. Soak the lid(s) overnight, fully covered, in a bowl with one or two tablets. Can get cheap own brand tablets at the supermarket. Leaves a slight minty taste so may want to rinse afterwards but not found this to be an issue.
You can buy a tiny brush for cleaning the colic vent of tommee tippee bottles - it's a bit like a pipe cleaner but a proper brush. I would think that would get underneath well. They sell them with the bottle cleaning brushes in supermarkets etc.