The problem is MRIs aren't there to "put your mind at rest". They're extremely expensive and extremely sought after; even most large hospitals will only have a single MRI machine. Not only that but a lot of primary care trusts can't actually refer straight to MRI, you'll have to be referred to a neuro /w special interest in migraines who may refer you.
Unless you have any red flag symptoms which highlight it could be something other than migraines it's unlikely you'll be given one. I don't want to come across as a dick but if you have clear migraine signs what do you want them to do? What do you expect to gain out the MRI? If it's clear you're having migraines and you have no other symptoms apart from they're severe and they've gone on for years it's highly unlikely you have something else. What other differential are you considering?
If you really want one but your doctor doesn't think you require one (you know, the guy who's trained for years and probably has more experience in medicine than you have in life) then go private. The waiting list will be huge for you on the NHS if you manage to persuade your doc to give you one because you're not a priority, regardless of how severe your migraines are. At best you might get a CT out of it.
I know this is not what you want to hear but I'm trying to give you a reality check, tests like MRIs are not there to give patients piece of mind. They are expensive, difficult to come by, difficult to refer to and in constant use. They're there as tools to diagnose, not give out for no reason.