You've been turned down for a visa, and now you intend to try and enter the UK visa free and get entry clearance at the airport. This might be possible, but will require excellent preparation. You're flying in four days time, so you'll need to work quickly.
First, expect to be delayed at Heathrow. This could be short (and they send you back) or it could be for some hours (and they might still send you back)
The Immigration Officers are concerned with two main things: can you support yourself while you're in Britain; and will you leave on time.
On the first point: $5000 is a good start. You need to be able to demonstrate that the money is yours. Think payslips and bank statements over a period of months. Money placed in your account by a friend to top up your funds wont be acceptable unless you can prove it's yours to use. If you hope to use a sponsor you will need a letter, and supporting documentation. If your sponsor claims an income of UKP20,000pa you will need something to support that, and so on. You will also need medical insurance to cover your stay.
On the second point, you need to demonstrate that you will return home at the end of your trip. This is especially important as they refused you on this. Concentrate on documentary evidence of ties to home: job, property ownership, family commitments, etc. For a fourteen-week stay this might prove difficult - most employers won't let their employees out for that long. If your boss has OK'ed this, get it in writing from him/her. In fact, get a letter from your employer detailing what they've agreed, regardless of the period.
Consider shortening your trip. It's a lot easier to cover, say, a four-week stay. Paying the airline to change the return ticket will likely be less painful than losing the whole trip.
Have a clear itinerary. Be clear about what you want to do, and when. If you intend to visit London, have a good idea of what you want to see and do. If you want to go hill-walking know which hills and travel with appropriate clothing.
Have your boyfriend and sponsor available and accessible, preferably in the airport, and prepared for a potentially long wait.
None of this is a silver bullet, and you may be refused anyway. The real solution is to apply for and get a visa. Two refusals will make it much more difficult to get a visa. If that happens you will certainly need a good lawyer to guide you through the process. Consider whether your approach is worth the risk, or whether you might be better abandoning this trip and dealing with the visa issue first.
Good luck.
