This kind of thing really gets on my nerves - partly because I have a dog with a really bad food allergy but also because through someone not thinking or respecting someone else's wishes, it causes owners so much grief.
However, what you've got to remember is that your neighbour probably doesn't 'get it' or fully understand the consequences of what she's doing. If it was me I would lay it on thick, exaggerate and lie a lot in this instance as I don't think pussy footing around is going to do you much good. I would say that you are having real problems with their health - upset stomachs, messing in the house, you've had to take them to the vet, they've started stealing food, bin raiding etc Explain that you are aware your neighbour is giving them treats and it's affecting them both badly and causing you an awful lot of problems, so she has to stop.
If she still doesn't... let me have her phone number
Good luck!
Hannah xxx