The Scooter motor is the standard 'go to' for a simple FW axe, 300-350W normally, and 500W if you are crazy. They are large and heavy but produce a lot of torque for very little power.

Check out Hatchet for a good example of a FW axe. I believe its a 5:1 ratio in 3/8in chain. We use a 4:1 ratio on our axe bot Matock but rather than a chain its a HTD T5 timing pulley profile.

Brushless axes are a thing, but they require a lot of gearing and a high end ESC to cope with short, high current loads that axes need to fire. They are also not that much more effective in combat, though they do leave slightly deeper dents.