It's a fixed blade so as far as the letter of the law goes, it's illegal.
No mate, ya got that wrong. They are not illegal. There is no law that mentions axes specifically, and fixed blade knives are not illegal either.
Okay, I know where I stand regarding carrying a knife, but where does the law stand regarding carying a small hatchet on your belt?
With regard to axes, the main thing to remember is that most of our laws are based on perception. You could take a knife or axe and in one circumstance, it's legal and in another circumstance, it'll illegal. If you are carrying a knife at your allotment to cut your cabbages, it's legal, if you are carrying a knife at your allotment to attack the guy with the massive runner beans, it's illegal. It's context and intent that determines legality and that's down to judgement. In the first instance, it's usually a copper who makes that judgement. If you are carrying an axe in the woods while collecting firewood in a benign and harmless manner, I would think that any copper would have no reason to be concerned in any way because you are not breaking any laws. But if you are running naked down the high street at 3am swinging it over your head, you'll have an armed response unit covering you in little red dots in no time, because your behaviour is against the law.
Axes, by their nature are more powerful and intimidating when used as a weapon, than a knife. It's understandable that in some circumstances, carrying an axe might cause eyebrows to raise before carrying a knife would. If you are sensitive to that and moderate your behaviour accordingly, and you have a reasonable reason for having one about you in the first place, you should have no problems carrying an axe of any size - in fact in some circumstances a really big axe might be more acceptable than an itty-bitty one. There are other things that affect perception as well as context and behaviour, such as appearance, age, police history etc.
It's all about the context. Use your judgement.
...so where were you thinking of carrying the axe on your belt? The woods at the back of your house, or when popping into Tesco's for a case of stella?