It would be an interesting experiment. The vast majority of wealth generation in London comes from financial services, those would suffer with an interruption of their connection to the rest of the UK before facing stiff competition as institutions set up shop elsewhere within Britain (the big one being the Bank of England).
The most interesting challenge I could forsee for the future overlords of London would be keeping trade alive while preventing financial flight, as well as brain a brain drain. It seems like the only way to achieve it would be a Soviet-style system of internal passports (and walls with death zones) where, even when individuals are allowed to leave, their families are kept hostage to ensure their return and a dual currency system similar to what Cuba employs, to permit large scale financial transactions while ensuring individual earnings aren't traded away elsewhere (either for goods that are hard to come by or for safe passage out of London).
Infrastructure would also be challenging, I wonder if the Underground and Crossrail could be adapted to keep Britain's rail system in tact (with trains not stopping, as they did in East Berlin) or if a railway along the route of the M25 would be required. On the Democratic People's Republic of London front, the Thames would require extensive alteration to accommodate Panamax ships and Britain would have enormous leverage since she'd still control the Thames Estuary (to say nothing of the Thames barrier, I wonder how much the DPRL would pay Britain to keep from flooding).