Sounds like you need a decent saw and an inexpensive hatchet with enough weight for splitting to me. Both will come in within budget and offer you far more flexibility not to mention cutting efficiency.
Example:
Bahco 396 Lap Laplander Folding Saw from Amazon at under £17 delivered.
It goes crossways through timber faster than any axe unless you're chopping like a maniac. It is light, packs well, and super-efficient at what it does, or a folding bow saw of some sort if you want to tackle bigger cuts would work. Both leave a neater cut as well which should be important to you if you want landowners to invite you back onto their land...
Any car boot fair or second hand shop will have a selection of hatchets to choose from or if you search on Amazon in the DIY & tools section for the word hatchet you will find lots of potential contenders.
If you absolutely must have an axe then try running a second search in Amazon's DIY & tools section for the keyword axe and you will find quite a few within your price range. None of them are great but some of them a fairly useful overall and it's an unusual oddity that many of the less expensive axes are ground in a fashion that makes them better splitters. You need to figure out whether you want to chop or split though, because all axes are a compromise and most won't do both of those tasks well.
Spear & Jackson - Razorsharp Razorsharp 2.5lb Axe at £18.99 could work if you dress the edge up a little. They're OK overall and have potential if you work on that edge a bit.
If you throw a few quid extra into the pot you open up options that include the Cold Steel Trail Boss which is (possibly too) big but very capable, or you might find one of the special offers on Husqvarna or Hultafors, or a used Gransfors Bruks may come on the market.
Recently Lurch was selling a Hultafors Trekking Axe for £40 delivered on this thread:
http://www.bushcraftuk.com/forum/showthread.php?t=65860 and it's a great general camp axe without going up into the silly weights and size.
To be honest I think you'd be much better off looking for an inexpensive eBay Elwell 2 1/2lb or smaller head and re-haft it yourself - there are a couple on eBay right now.
I'd strongly suggest a saw over the axe though - faster, neater, less effort and less weight on your back.