Gail, I'm assuming that this ties in with your other post about Edinburgh as well so haven't responded to that one.
I totally agree with Harter's comment. If the items are not sourced from a central preperation area then the very nature of the restaurant is likely to have the risk of exposure to peanuts due to the type of food. By this, I mean that the asian food options will be at the cheaper end of the market, but the kitchens will often use nuts in some of the cooking so there is a risk involved.
Best I can sugest is some of the Italian places which aren't part of a chain. I'm not a huge fan but Vitoria's springs to mind and their pizzeria on Leith Walk is decent. There are also a couple fo Mexican places that I'd presume were suitable. One on Leith Walk and one on Lothian Road, let me know if you wnat more details
The difficulty you are going to fnd, particularly at the cheaper end of the market, is somewhere that will guarantee nut free. It's because at that price point they are buying in so much from a variety of suppliers that they do not have sufficient control of the preparation process. In fact, you will find many restaurants offer the disclaimer that they cannot guarnatee nut free.
There may be some useful info here: