The fact your onward destination is to somewhere in the Schengen area is essentially irrelevant.
(Given that the UK is not part of Schengen, the only relevance is to whether UK immigration consider you would have any obstacle to entry to where you are going, and what potential there might be for you to be denied entry / refused travel to there).

So one question would likely become can you satisfy them you qualify for entry to that Schengen area country - do you have the right visas/passport types, etc.

To answer your specific question:

Firstly, changing 'city' of airport is in itself perfectly fine, indeed given the catchment area of many airports overlaps I would not get concerned about what city name it prefixes itself with..
(e.g. Birmingham airport is virtually as easy to transit to in London compared with transiting to any 'London' airports').

Indeed, unless an airport is setup to provide international airside connections (which only a limited number in the UK are) then it makes little difference whether you are transiting between flights in the same terminal, different terminals, or different airports.

However, I would expect to be asked (given the abundance of route links between Europe and North America) why you have chosen this route - both going via the UK and also with a distance between connection points). And I'd expect to be asked how you plan to travel between Manchester airport and your London departure airport (don't dismiss the distance between the two).

Indeed, personally, I'm also curious about this.
Even if traveling as cheaply as possible in economy, I can't believe that this will be cheaper for you, or less hassle. Especially doing it both directions.