You'll want one of these (all are HD, don't buy a non-HD projector):
InFocus SP5000 - £666 - £850. About the brightest you can get, but the lense is fixed and the contrast is below average for this generation of projectors.
Panasonic AE700 - £950. Previous generation, but the best of that generation. Its not paticularly bright so you'll need a dark room. Very good contrast and a strong feature set make it a good candidate.
Panasonic AE900 - £1024 - £1100. Very, very good colour and contrast. Still not overly bright though. Has SCART input which is a definite plus. Best featureset of any projector listed here by a large margin.
Sanyo Z4 - £1100? The arch-rival of the AE900. It doesn't have as good colour, but it may have slightly better contrast and is certainly significantly sharper. You will see the LCD pixel layout if you're too close.
Hitachi PJ-TX200 - £1100? A very good projector, imo better than the Sanyo and AE900. However, it's inputs other than DVI/HDMI/VGA are poor. Its not good for PAL or even general TV. Scaling is bad, as is de-interlacing.
You want to use component with an Xbox 360, it works better with older Xbox games. You will also need to use a proper screen unless you go for the Infocus SP5000 which is so bright you don't need a screen

Also you'll need to replace the bulb every 1-3 years (£100-£300 depending on the projector).