I’m from a similar area to you, so I know what it’s like to be a after a local base.
There are plenty of airlines in your catchment area, but the problem is that most don’t take newly qualified pilots. The best chance you’ve got of staying local is the Jet2 apprentice scheme, flyBe, or Ryanair (providing they start recruiting again).
LBA have Jet2 and Ryanair. The Jet2 apprentice scheme is in my opinion the best crack at getting a local base. As far as I know Ryanair aren’t recruiting at the moment, and are looking to downsize their Leeds operations. Eastern Airways still do a few flights out of there, who take new hires.
MAN have flyBe, Jet2, Ryanair and easyJet. To my knowledge, Manchester is one of flyBe’s most-wanted bases. Lots of internals want it, and the odds of a new hire getting it are slim. Same problem as above with Ryanair, and the main way into easyJet is through an integrated school with a £40k type rating. TUI, Thomas Cook, and Virgin don’t take newly-qualifieds.
EMA have Ryanair, Jet2, and West Atlantic. West Atlantic could be an option, most probably on the BAE ATP on nights. Same problem as above with TUI.
LPL have Ryanair and easyJet. Same issues as above.
DSA sadly are waving goodbye to flyBe, and are left with only a TUI base.
BHX have flyBe and Ryanair. It’s probably a decent chance of getting a flyBe base there.
All in all, when you get your licence it’s needs and musts, and you’ll probably be grateful of any offer that comes your way. It’s not unrealistic to expect a local base. But, it’s much better to expect and prepare to relocate. Even the airlines with bases close to you are perfectly capable of sending you to far flung corners of the country (or even Europe in the case of Ryanair).