If we're going English it'll be Howe or Potter. Not sure I want either and I doubt they'd be any better than Southgate. Whether you like his football or not, there's no doubt that he's managed to get us closer to winning than anyone else in almost 60 years. And all while maintaining his dignity while being abused and vilified for not playing like 1970 era Brazil and narrowly failing to win a trophy. If we're going overseas I'd go Klopp or Emery. Not sure either of them would think the job was worthwhile though. No-one ever seems to leave the job with their reputation in tact nevermind enhanced.
I used to be naive about this (back in the 80's & 90's), where I always hoped it would be an English man, who got and did the job. Which I admit sounds ridiculous these days, when you think about it. Yeah we had the Sven Eriksson and the awful Fabio Capello era's, where they both couldn't get us beyond the Qtrs in tournaments.