I don't wish to throw cold water, but I think the legal situation should be the first consideration before you contemplate using an axe, and certainly fitting it visibly on your bike would be ill advised. Where I live, in Ireland, you can go to gaol for a year and be fined heavily for carrying a pocket knife without a reason. In the UK, you can be gaoled and fined for carrying a knife without a good reason, and for carrying any cutting instrument over 3 inches; most pocket knives now sold in the UK are "UK EDC friendly", meaning no blade over 3 inches. A lot of places in the world an axe would be ipso facto evidence of malign intent.
I met a guy on the bus where I was hovering and havering with my pencil over my sketch of a woman and child in the row before me. He leaned over to sketch the line I wanted in the air, so I gave him the sketchbook and the pencil and he drew just the line I wanted. We struck up a conversation over my sketch kit, housed in ostrich skin, with a custom reconstruction of a 1930's clutch holder for really thick leads, and it turned out he is a well-known artist. While I was scraping a pencil with my penknife, he told me a horrid story. He's a bit scruffy looking because he doesn't have a base and travels where the notion takes him there'll be something worth painting. He was stopped in the street in England and searched by the police, and flung into a cell for having a penknife, despite the plethora of painting gear in his luggage and his protests that he needs the penknife to sharpen his pencils. When he asked to be taken to a bathroom so he could brush his teeth, he was told inmates can wash and brush their teeth in the toilet in the cell. He was there the entire weekend. On Monday the magistrate, a less cynical and more sophisticated type than the pudding hats, and less malicious too, asked for a demonstration of how he sharpens a pencil, apologised for the over-zeal of the police, and let him go. I resolved there and then to move my tiny penknife -- less than 2 inches -- from my keychain to my painting bag instead. (Which is a PITA because the wee thing gets lost and it's tiresome finding the right shape and flex of blade that won't break leads under its own weight, and expensive too.)
In these days when the police consider you guilty of a thought-crime even if they have no reason to consider you bear malicious intent -- which is what these knife laws come down to -- it's smarter to give them no reason to make some far-fetched case against you rather than to depend on their good sense and goodwill.
And that comes to you from an Irish scofflaw...