There are quite a few threads on this site (mainly in the private flying section) which explain in some detail how to do this.
It's actually very easy. That's the good news.
The bad news is that if you're off to the States next week, you've left it too late
Briefly you need to send a form to the CAA with some money. You then send off more forms to the FAA. The CAA contacts the FAA, who send you a letter within a few weeks telling you to make an appointment to collect your temporary FAA licence.
You won't be able to do all this by next week though.