For my tuppence worth I would suggest getting the theory done over the winter months and then taking time out to get your licence when the lump sum becomes available. Depending on when and where this could be done in 4 weeks, or so, and if you chose to go abroad it could be done during the winter months. With a lump sum coming I would opt for this approach.
Regarding weather; generally speaking it is surprising how bad the weather can appear and still be flyable. There are plenty exercises that only require the weather to behave in the local training area for the time of your lesson. The key is having these slots booked, or being available at short notice. Once you get to the nav exercises and post skills test then the weather further afield has more influence on your training/flying.
Flying with a total overcast is not an issue, nor is flying when there are rain showers which can be flown around. A lot depends on where you fly, Prestwick to Fife with a 3000ft ceiling should be easy enough, Prestwick to Inverness with the same ceiling poses a few more challenges. Localised showers are relatively easy to see and avoid although more care is required in hilly terrain where the escape routes are not as numerous. Of course you also have to consider the airspace around your route. To fly in the Lake District clearly requires a higher ceiling than flying in East Anglia .
For me the big challenge is assessing if what is flyable weather now is going to remain so, is the ceiling going to lower, are the localised showers going to become more widespread?
What I say above is not intended to give the impression that weather should be taken lightly, it is intended to shed some light on your question in explaining clear blue skies as far as the eye can see are not essential for VFR flight.
Best of luck with it.
BB