I built a Westfield 7 years ago and can vouch for the fact that it's a great experience. I didn't bite off as much as you have, of course. I started with a bare chassis but getting that far from scratch would have been 3 times the work of the rest of the build - for me at least.
I would echo Mark's comments about the "under a grand". At the end of the day if you built the whole lot out of scrap you could maybe do it (then again, if you want it road legal the best part of 300 quid's gone on the SVA test) but it's a balance between time and money at the end of the day. If you try and restrict the budget too much you will get fed up with it before you're starting to see the light at the end of the tunnel. There will be a lot of work turning scrap parts into something you can feel proud of. You could make something roadworthy but not pretty with considerably less work but it'll still be a real struggle. Then again, it's a balance you can adjust along the build as time and finances allow.
I agree with Mark regarding the engine choice too. A single seater would be a no-brainer. Bike power all the way IMO. If it ends up as a 2 seater bike engines are a love-hate thing. Many people swear by them but you need to keep the car as light as possible and can't afford to include any creature comforts. They are a pig to drive on the road in stop-start traffic, especially if you don't have a reverse gear, but once you get them rolling you have a 6 speed sequential box and 12,000 RPM red line to play with
It also depends on the proportion of road / track mileage it will be doing, of course.
I went for a 2.0 Ford Zetec which was probably the right choice at the time although sometimes I wish I'd gone for a C20XE instead. (These days the Ford Duratec or Audi 1.8T would be high on my list.) It's well mannered on the road, has plenty of power and doesn't add too much weight to the car. Mine's tuned and is probably making 180-200 BHP, although it's never been on the rollers (it was mapped on the road using a wideband O2 sensor and lots of datalogging).
There are also those who insist you need a V8 which is far too much engine for a small car IMO. Weight that will make the car much less nimble, gobs of torque where you least need it in a light car and an engine that doesn't like revs. The sound, though...
I wouldn't be too hasty to knock Ford underpinnings. Whatever your opinion of the cars, the RWD mechanical parts (engines, gearboxes, diffs) are bullet proof and used throughout the kit car industry. There's much more support for doing unconventional things with Ford bits than anything else, IMO. Having said that an Omega would make a fine donor if you want to be different.
Feel free to drop me a PM if you want to bounce any ideas off me. It's very easy to set off on one track and end up with not quite the car you wanted in the end so get as many ideas from others who have trodden the path as you can, blag as many rides in as many cars as you can, and think about whether you'll get your greatest fun from doing track days or commuting in something a bit "different".
Kevin