Fitting in platforms like this would certainly be more challenging than just rebuilding the existing platforms, but that doesn't necessarily mean it's impossible (and if only certain stations are tricky then one or the other service could skip them).
The Watford DC shared platforms are I believe compromise height - it's certainly a step down into a Bakerloo line train and a step-up into a London Overground train. Tube-gauge trains run past sub-surface height platforms all the time without any issues (battery locos are tube gauge and go all-over the system, the 1938 stock train semi-regularly runs all-over the sub-surface network, including recently to Amersham, etc past platforms that are also served by Chiltern services), Piccadilly line trains occasionally run over the District lines between Acton and Barons Court, and very occasionally get diverted into Ealing Broadway District line platforms.