Where would the “LO” service go to the south? Given that the TL core is full and St Pancras is full, and Belsize tunnels are 4 track.
Well, the easy option is to just terminate at West Hampstead, where passengers can interchange to Thameslink to continue south, or head down the road to the Jubilee or the existing LO route between Stratford & Willesden (would be nice if the Met stop was moved there from Finchley Road too!). In time there's a likelihood you will also have the option of running down the Dudding Hill line to OOC Lane and on to Richmond instead, but as you would miss out on West Hampstead, I can't see that being too popular. Perhaps St Albans to West Hampstead, then reverse and run to Richmond?
The more "ambitious" option for me is another one of my pet concepts of a new tunnel from the Finchley Road area alongside the current MML tunnel portals to the Goblin somewhere around Tottenham North curve tunnel #2, potentially severing the current rail link there by removing Carlton Rd junction. That new tunnel you can then make as much or as little out of as desired, i.e. optionally adding underground platforms at Gospel Oak and Belsize Park LUL, or indeed, somewhere new. That doesn't get you any better interchanges beyond West Hampstead heading south, but it gives services a useful connectivity function by giving the Goblin an outlet, though admittedly the bigger benefit would be to freight by giving it a path from the Goblin that didn't need to interact with the Thameslink tracks at all unless heading north of wherever the Hendon lines end up finishing. I believe something similar popped up in Boris's ill-fated R25 materials?
But anyway. Let's focus on the MML aspects. With a base assumption that you could run a frequent segregated LO service between St. Albans and West Hampstead, and Thameslink could run fast between St. Albans and West Hampstead, would that free up sufficient capacity whilst still being acceptable for the good people of Radlett, Elstree & Borehamwood, Mill Hill Broadway, Hendon, Brent Cross, & Cricklewood having to change onto Thameslink services at West Hampstead, given that there could end up being more frequent Thameslink services if capacity were able to increase?
Additionally, could there be a case for opening new local stations served by the LO lines, i.e. midway between Cricklewood and West Hampstead, a Northern Line interchange at Colindeep Lane, Grahame Park, somewhere between Scratchwood and Apex Corner, Napsbury? ...or would these stations make the service too slow? Could that be mitigated by an additional Thameslink stop at, say, Mill Hill? Obviously there is still the issue of Kentish Town, but at least there are four platforms there, so stopping Thameslinks could be overtaken.