I have funding to start my PhD in September, but would like to wait until after Christmas. Has anyone any experience with these issues? Would it be considered very rude to ask for something like this?
The reason that I want to wait is that I would like to finish off the project that I am currently doing - don't want somebody else to take over my project and possibly publishing it, when I did most of the thinking...

In funded projects it's usually not possible - the project usually has a start date which is the start date for funding as well, and I suppose it's pretty complicated to change it especially if there are several entities involved (eg. University, Department, Research Council, etc.)

If you are self-funded it shouldn't be a problem. In fact some universities have 2 start dates for PhDs - one in September/October, one in January. If you are funded externally (for example by a charity, by your employer...) I suppose there won't be any issues on the university side but you will have to talk to your sponsor to clarify the start date of the scholarship.

Nimrod's right when it comes to external funding it's normally non-negotiable except in exceptional circumstances (IE pregnancy, illness, etc.

My uni actually has 4 start dates - in Oct, Dec, April and July on the 1sts of each month. I wasn't offered my (externally funded) PhD studentship until 10th October 2002, I wasn't even interviewed until about 6th - in otherwords, past the Oct registration date. I eventually got started by end Oct, but my registration was backdated to the 1st, rather than commencing the next available date. This was becasue of funding conditions. It effectively meant that I (officially) has a month less that everyone else.

Ok, thanks for your replies.
I have been offered a university studentship, so at least it is all within the university. I suppose I'll just have to phone up my supervisor and ask if it is at all possible... Do you think he might be offended or put off by this? I mean, I really do want to do my PhD with him, but ideally I would like to finish this first.

I initially applied for start in October but once I was admitted I asked my supervisor and the admissions tutor if I could defer my entry until January (I was working at the time and my employer had asked me if I could stay longer). They didn't object at all, but at that time I hadn't been awarded funding so I was planning to go self-funded. Later on I did receive funding from my university but one of the requirements of my scholarship was that I had to start in October. I was so impatient to leave my job and so happy about my scholarship that I just started in October. However I don't know whether I could have convinced the University to let me start in January and still keep my scholarship. I mean, for the Department it probably doesn't make a big difference if you start now or in 3 months (unless they are planning to give you some teaching or other duties) but the administration is usually more rigid and they have clear procedures. Anyway I wish you luck.

I did exactly teh same - and I have external funding. My course would have started in September but I was working at a good project and asked whether it is possible to defer. It was no trouble at all as my supervisors saw the point that I am not asking for a holiday but want to finish my project and get the publication out. Just ask!