Entirely do-able within the UK system; you'd need to get CAA on-board from the start and it would be Annex II so exempt from any EASA involvement.
If you design a new aeroplane that happens to look like a Mosquito that would be quite different to a true replica.
The new aeroplane would probably need certifying through LAA to an agreed subset of CS.23 and would be no different to designing any other homebuilt through them. Broadly, this is what has happened with the Australian 70% Spitfire replicas we've been seeing in recent years.
The replica would probably need certifying to PtF directly through projects at Gatwick - you'd get away with something less compliant with modern standards, but you'd have to go to great lengths to show it matched the original design as far as possible. This has been the case with most WW1 replicas.
Either way, build control, inspection and testing through build, then flight testing after build would be pretty large jobs. But it's been done before and doubtless can and will be done again.
Sort of thing I have been known to do for a living in the past (the certification and testing, not the design and build, but it all bundles together). If you've got half a million or so to spare, and fancy luring me away from my day job, I'm open to offers.
G