This would be an excellent computer for your intentions.
http://www.amazon.co.uk/gp/registry/wis ... av_lists_1
The total is £473.52 from Amazon.co.uk.
For wireless internet you could get a USB WiFi dongle. I would also suggest a USB hub for your controllers, if you trip over the cable you won't break the case.
edit *I previously had a micro-atx motherboard that would not fit the case on the list, I swapped it out for a micro-itx board so everything should be fine. Thanks for pointing it out, always double-check the parts when people like me suggest a build*
It may be more than you wanted to spend, it could be adjusted.
I picked high quality parts. High quality doesn't always mean high end, or unnecessarily expensive. They are well designed, durable and come with excellent warranties (Corsair).
It would also be capable of a lot more than MAME, it would handle any kind of emulation and even recent PC games.
If you intend to use GroovyMame to connect the machine to a CRT you would want a GPU compliant with that.
Turns out intel i3 processors Can't be overclocked, but this one runs at 3.7Ghz which should have no problem running SH3 at 100%, modern chips can do a lot more with their clock rate. An equivalent clocked i5 quad core would cost a lot more.
Your local Amazon will have all of this, as will any good retailer of computer components.
Even though I am almost certain this combination would be compatible, It would be worth going over the list carefully and double checking that everything is right.
*see above edit XD
This is what I would get for a dedicated emulation box.
You could even drop the GPU as the processor has on board graphics. Reduces the price to £393.52. The machine should run MAME just as well. Having the GPU I listed would be for emulation that utilizes it, and PC games.