The Sword of Wounding states that all damage dealt with the sword can only be healed by a short or long rest. If a rogue uses sneak attack with this weapon would the sneak damage be included under the sword properties or is this considered a different source of damage?
While not explicitly spelled out, it's generally accepted that Sneak Attack Damage is part of your Weapon Damage, the same way your Stat bonus to damage is part of your weapon damage.
It is treated as being the same Type of damage as the weapon you are using deals, and it benefits from all of the other things that impact damage.
Based on that, yes...a Sneak Attack done with a Sword of Wounding (assuming said sword was a Finesse weapon) would apply the Sword of Wounding's effects to the entire damage stack.
This is further supported by developer commentary. This is from Mike Mearls, so it is not authoritative, as if it came from Jeremy Crawford (the 'official voice' for WotC rulings), but it does still represent designer intent.
Q: If sneak attack is used, are sneak damage dice part of the weapon damage dice? are sneak dice maximzd with sharpness sword?
A: i'd say yes
According to Sage Advice, sneak attack damage dice are considered part of the weapon's damage dice:
Svargth @Svargth @mikemearls If sneak attack is used, are sneak damage dice part of the weapon damage dice? are sneak dice maximzd with sharpness sword?
Mike Mearls @mikemearls @Svargth i'd say yes
So yes, it probably would be included, and would require a rest to recover!