Right, fixed it. It acts differently between the Gear 1 and the Gear S. I've turned off notifications on the Messaging app on my phone and, on the gear 1, the messages are still forwarded to the phone. However, this isn't the case on the Gear S so I've had to re-enable notifcations in the messaging app.
That fixed the issue. However, now I've figured out that if you turn on the Mobile signal permanently on the watch you may as well just get the twitter alerts being sent to the watch's SIM number rather than the phone so I'm doing that. Working a treat!