Hello Ben - welcome to the forum, and the wonderful world of getting splinters - er, I mean making stuff from wood!
Buying hardwoods in small quantities is actually quite difficult. Almost all 'local' timber merchants are set up to cater for the building trade or bulk users, and sadly some regard the small buyer and the amateur as a damn nuisance. Consequently, you have to either travel a bit, or seach out the mail order suppliers; this usually means paying a premium price.
That said, try Roger Haydock in Widnes. They may not be keen to machine and deliver small quantiies, but may be more amenable to machining up small quantities for you to collect. You can but ask, and their stock range is quite good.
Of the mail order suppliers, try Interesting Timbers is Somerset, and British Hardwoods in Yorkshire. Both will supply PAR hardwoods, though not always in all species.
Another option - and one I'd urge you to consider - is to learn how to plane up your own timber. I fully appreciate that this sounds a daunting prospect for a relative newcomer, but I promise you that you won't regret investing the time to learn. It allows you to use sawn timber (cheaper than planed!), and allows you to salvage good timber from junked items - often free. You don't necessarily need a lot of kit, and it's a skill you'll need to develop anyway if you're to make the most of woodworking. Robert Wearing's book 'The Essential Woodworker' (Classic Hand Tools or Axminster Power Tool Centre can supply - Amazon can't) is very good on planing basics, and much else.