Minecraft isn't very optimized, but optifine really helps.
Texture quality is limited by VRAM (video RAM), so the more RAM you have on your video card, the higher texture quality you can have (that lucrative x512 sphax texture will run you a minimum 2-3 GB VRAM). I run MC just fine with x64 sphax with a sandy bridge Pentium, Radeon 6870, and 4 GB of RAM. Which was a budget build of mine and came with a price tag of less than $500 (£310). That was in Jan/Feb, so the prices have dropped on some of the stuff. You can probably pull your HDD and still use that, so that'll save you a good chunk of money (I spent $120, or £74, on my SSD -- well worth the money but still a good chunk of change).
Not having a SSD will affect your performance, since HDDs are terribly slower. This will affect your chunk rendering speeds and loading times, but if you want to be cheap, then we can do that.
First thing to do is select a mobo. I'd definitely suggest and Intel chipset, since you're going to want a discreet graphics card there's no reason to get any of AMD's stuff. H61 and H67 are great choices.
MSI H61 - $70 (£43)
MSI H67 - $90, $75 after rebate (£56, £47 after rebate)
If you're really scrounging then go for the H61, but if you don't mind rebates (gotta stay on top of it too), then you can get H67 for a bit more which is only a bit nicer, but well worth the price increase (not required for the build).
Next we need a processor to fit in that LGA1155 slot. Which isn't too hard, there's lots of great ones available for a pretty cheap price. The choice is pretty clear, a sandy bridge Pentium is pretty powerful for its price, and fits great in a budget machine (it's way faster than the core 2 duo you have now, that's small potatoes compared to the newer sandy bridge architecture, don't let the clock speeds fool you). The i3 is about 2x more expensive, and you don't really need it to run minecraft well, so just skip it.
Intel Sandy Bridge G630 Pentium - $65 (£40)
You're definitely going to need a new PSU, your 250W one will sputter out like a solar panel trying to run a miner. The Corsair CX 430 is a nice deal, even has a rebate. If you can get it on sale it's less than $20 (£12). I definitely advocate waiting until you can get it on sale, since you're scrounging. It does make a bit of whining noise, but it's not loud and I can't even hear it over the low hum of my fan, pretty quiet noise, nothing to worry about (and you won't find anything else like it for the same price).
CX 430 - $45, $35 after rebate (£28, £22 after rebate)
A graphics card is really important, and you won't be able to get a new spangled one, but an older model should do nicely. A radeon 6870 does very nicely for it's price, but you can go down to the 6850 if you want to pay less (I don't think it's as worth it) or go up if you want to pay more for more or less power.
XFX Radeon 6870 - $180, $150 after rebate (£112, £93 after rebate)
RAM is pretty cheap, and doesn't really matter what kind you grab, but I do prefer Mushkin. The price isn't really different from anything else you find (you can get some for pretty cheap). RAM is limited by your OS, if you have x86 you're only going to be able to stick in 4 GB (enough for me), but if you have x64 you can pretty much have as much as you want. But more on that later.
Mushkin DDR3 4GB - $19 (£12)
Total: $324 - $399 (£201 - £248)
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And that's pretty much all you'd need, I'd like to see what kind of case you have, if any, to see if we have to get a new one or we can use the existing one. This build is pretty similar to my own (a smidgen more powerful, though). I'd hate to compromise on anything to bring down the price, because it would just sacrifice a lot of power for very little money. You could probably do an AMD build and come up with a lot less without a discreet card, but if you want that x64 sphax and all of the goodies, this is what it will take.
You're fine with vista, just keep your stuff clean. If you want 8 GB it's going to cost extra, and you would need x64, which you probably don't have, running you a good sum for a new windows 7 installation. You won't have to reinstall or anything, just plug your old SATA HDD into the new mobo and it'll boot up with all your old stuff on it.
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I don't know what the vista hate is all about. 7 is better, yes, but for a guy that only wants to spend £180 on what needs to be a new computer you sure ask a lot. Especially when there isn't really anything wrong with vista. You can turn off the aero theme and it's not as terrible performance wise as people have been led to believe. All of the permissions can be edited in the control panel if you don't like being asked permission for everything.