You can complete a full PPL in 3.5 weeks. This is Fact. If you can stump up the money, it will be by far the cheapest way to do it. Then it's done and the license is in your pocket.
Find yourself a flight school in the country, ring them up and tell them you'd like to do a PPL across a 3 week period and talk to them about what that might cost you and what it might look like.
Personal tuition, fly everyday, better weather and most likely 1 or 2 instructors for the whole process.
You won't find this combination at any school at YSBK. This is why some of the bigger schools have setup satellite flight training bases in NSW/ Aus.
PM me if you'd like some personal recommendations from NSW and yes there are plenty to choose from. They are honest, no B/S small business owners and can offer a better solution than the city folk can if for no other reason but the weather. No offense intended.
I know an Australian bloke who worked fulltime in London who had always been in and around Aviation but never learnt to fly. He did his full PPL in 4 weeks in rural NSW and that included spending time with family. From zero hours to PPL. He was 38 at the time and this was around 2008.
You don't need luck, you need cash, time and a priority list. If a PPL is at the top of that list, then make each day count towards it until it's in your pocket.