The Cadian Regiment grants Common Lore (Imperial Guard) and Common Lore (War).

The Weapons Specialist skills are Common Lore (Imperial Guard) and Common Lore (War).

Does this mean that a Cadian Weapons Specialist has Common Lore (Imperial Guard, War) at +10 each, or just at the Known level, and the skills do not stack?

Yep, they stack, so they'd both go to +10.

There's a CO type (Choleric?) that will add those again if you want +20 to start.