I know most people here don't like them, but I think either of the following would make really a really good England manager:
(1) Alan Pardew
(2) Harry Redknapp
Being a good international manager isn't really about training a team Klopp-style or buying youngsters a la Wenger.
It's about putting together a solid unit from the bits and pieces available to you, having a good tactical sense and a game plan that isn't pretty but works, and instilling an intense but unsustainable team spirit / siege mentality over the course of a tournament.
It doesn't matter if you are a total asshole who drives everyone crazy after a few months.
Rafa would be great too!
Pardew, apart from being an @rsehole of the highest order, is a VERY streaky manager indeed. His teams always have a spell of looking unbeatable followed by a longer spell of being unspeakable, rinse and repeat. I think he'd be a disaster.
'Arry though isn't a bad shout IMO, perhaps in a DOF-type post with a younger first team coach type. Eddie Howe might be a possibility for the latter role but I just get this sense with him that he's so attached to Bournemouth that he might not transplant well. I think Rafa could be the best shout of the three you've mentioned, but (a) he's given Newcastle a commitment which it would cost shedloads to release him from and (b) I wonder if the Spain job will fall vacant soon.