You don't say how you plan to travel, which makes this request a bit difficult to place in context. Are you a backpacker who wants to know about cooking in wilderness areas? Are you driving around the country and want to know about cooking in roadside areas? Are you planning on arriving in town centres by train or bus, and want to know about urban areas?
My first thought is that wherever you can BBQ in public, you should be both legally and contextually fine. As far as I can tell, the law in England and Wales doesn't distinguish between full-on open fires, BBQs, and portable stoves; all make heat by burning fuel, and are treated similarly. The good news is there seems to be no national ban on BBQing in public. The bad news is that local councils do have delegated powers to forbid open fires in areas under their control, and everywhere in England and Wales is under control of such an authority, turning the answer into "it's fine unless it's not".
If you call the local tourist information office and ask for public BBQ areas, I can see no problem with arriving at one of those and stove-cooking. A lot of them tend to be beaches, so sticking to the edge of the country should help. Many of the rest are in remote areas. My guess, and it's only that, is that if you set up in a layby on a remote rural road you should also be fine, because the chances of discovery are minuscule.
On private land, things are different, and the landowner has discretion. Some campsites permit open fires, and any campsite that permits tent camping must expect people to stove cook (I've never had a problem at a private tenting site, even those that don't permit open fires).