Did someone say execution axe?
Nah,it's just a common French axe,gents

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https://imgur.com/a/F1PYm8Q
Most shown above are clearly identifiable as woodworking tools....
Moonw,thank you for these interesting links(i can't watch them alas).Kevin was right in that the executioners belonged to a hereditary professional clans,things being socially awkward for them and their children et c.
There's much written about the history of those,and though i've never studied it specifically,i've never heard that any special implement was developed for when they did use an axe.
That one in the first video still is probably a hewing axe,or possibly a butcher's...(in Germany the developement of both happened in parallel,and some resemble one another).
I butcher a lot of large game animals,and can't imagine that a human neck vertebrae is much of a challenge for any decent sharp axe,maybe a straight-ish/long-ish edge would help those challenged in their aim...So any hewing axe* would do(butcher axes are also quite thin-bladed).
I remember vaguely some historic data how Peter the Great in the end of 17th c. on a sadistic whim made his aristocracy participate physically in the very many beheadings that they themselves decreed...These pampered wealthy,having never handled any tools,were awful about getting it done in one,or even two or more blows...The axes provided were of the most inappropriate kind,to make it all more fun...As a result a custom developed where the condemned could bribe the authorities,and for a large sum a real peasant was appointed,who could get their head off in one clean whack...(if ever one wonders how come RF is such a twisted place today

But nowhere that i,personally,came across were there any mention of a specialised human-neck axe...too easy of a cut to justify tool modification,i'd guess.
*part of the punishment was the Thrill,of course,the Theatrics...and broadaxes in general Do look impressive...thus this custom of today ascribing them that alleged duty,so maybe same went for those times,and the crowds loved them for that spectacular looks....