I'd go with Manchester. Living in Altrincham there's plenty going on locally but a strong connection to Manchester, via Metrolink etc., probably more so than if you live in Warwick/Stratford and want to head into Birmingham. I reckon that would give you something as close to a "London lifestyle" as you would find outside London itself.
Others are right that neither Manchester nor Birmingham has anything approaching the cultural life of London. But it's a bit much to suggest you'll have exhausted the possibilities in a few weeks/months.
For example, if you're in Manchester, there may be far fewer theatres than you'd find in London but they don't tend to keep the same productions running for extended periods as many West End theatres do. By the time you've "done everything", there'll be a new-to-Manchester touring production at the Palace or Opera House, new plays at the Library or Royal Exchange, etc. Similarly, just because you saw a concert at the Bridgwater Hall last month doesn't mean you've "done" there and it holds no further appeal. Different performers are performing different works now.
Also, who wants to go somewhere "new" every weekend? Is this a peculiar London attitude that you can only go somewhere once? If you like a particular restaurant, bar, sports venue, activity, is it so odd to suggest you might want to experience it twice? Why wouldn't you want to try new dishes at your favourite restaurant, or hang out with friends in a favourite coffee shop or bar?