Having synergies between weapon skills is a great idea, and I think even more can be drawn from it.

Every weapon type could influence 2-3 other types closest to it in nature, having a stronger synergy with some and weaker with another (I'm thinking either two strong ones or three weak ones as a rule).

My idea for synergies is based on a [highly hypothetical] scenario of ease of a person proficient in one type of weapons learning another, but you can well have something else on your mind.

For example:
Unarmed combat should influence daggers & knives (strong) and thrown (strong).
Daggers & knives should influence one-handed swords (weak), unarmed combat (weak), and thrown weapons (weak).
One-handed blunt weapons should influence chained weapons/flails (strong) and two-handed blunt weapons (strong).

And so on. Will gladly (re)balance every type if needed.