I am afraid you will be disappointed unless you alter your criteria. The number of one handed opening UK legal EDC knives is limited, and when you narrow down to thumb studs rather than holes, divots and spine jimping, you have only a couple, and they are more than £35 and don't have wood handles.
I would think that you also want something that is well made and takes and holds a good edge. If so, you can have a Boker Techtool for under £35, but no wood unless you make them yourself, and not one-handed opening.
The list that TeeDee posted at Heinnie is pretty much the full range of stuff that you could choose from. If you can't see anything you like on there I doubt that you will get any better offers from folk here. £35 isn't a lot for a folding knife, and you want a lot for the money.
I used to carry a Spyderco UKPK, S30V/G10 version, which was one handed. It got lost somewhere and now I carry the Boker Techtool1, 12C27/G10 and have not really missed the lack of one-handed operation all that badly. For me, in a legal EDC, it isn't all that important any more. I liked the convenience of the Spyderco, but if I am honest, I didn't need that few fractions of a second faster opening in order to open mail etc. Wood is pretty, but for me the cheap wood just looks cheap and I would rather have harder wearing, more stable laminated material (G10 or Micarta) than plain brown wood.
Best of luck.