There's no way to know for certain, today, but I believe the answer to this question lies in the fundamental modern misunderstanding which people commonly have with regards to the PURPOSE of Hadrian's Wall. Modern scholars no longer believe that the Wall was REALLY intended to be the massive defensive barrier that an average person looking at it tends to conclude. Even by Hadrian's time, the Romans would have been well aware of the futility of building one big wall to keep people out of certain area. If you can't constantly man the entire length of the wall (which would have been impossible even at the height of Imperial manpower on the island), then there will ALWAYS be gaps in the defense over which people can cross (with simple ladders or ropes.) The Wall was never meant to be the be-all-end-all barrier to raiding from the North. We see confirmation of that in the fact that the wall was never uniform in construction. Beginning as a broad stone wall in the East (with true defensive capabilities), it grows ever narrower as it stretches into the West (beyond the more settled, 'Romanized' parts of the province), until it turns into a fairly undefensive turf wall as it nears the Western Coast. A turf wall is little more than a 'keep out' sign to determined invaders.

Modern scholars believe that Hadrian's Wall (much like his walls and fortifications on frontiers around the empire) was intended less to be an impregnable defensive barrier and more to serve the dual functions of helping to CONTROL the peaceful movement of both goods and people back and forth across the frontier (likely for taxing and regulatory purposes), as well as to provide a fortified forward platform from which Roman forces could strike ACROSS the frontier when and as needed. Roman military policy even in Hadrian's day never really embraced the idea of cowering behind walls (it wouldn't be until after Julian's time that this really began to change.) The Romans preferred to march out and fight their foes in the field, on enemy territory. Hadrian's Wall essentially gave the legions stationed in Britain a massive fortified zone from which they could strike (and behind which they could withdraw to safety after their attacks) at foes in the northern part of the island, but it was never intended to be a massive defensive barrier that would stop a full-scale invasion. That's not really how such barriers work in practice, and by that point in history (after 200 years of stationary frontiers along the Rhine and Danube) the Romans would have been well aware of that. In this sense, the river Tyne (as with the more famous Rhine and Danube) more than adequately served the purpose of allowing the Romans to control the flow of trade and people back and forth across the frontier and of providing a fortified (but not absolute) barrier beyond which the Romans could strike when they needed to and withdraw back across when they were done--so there was simply no need for the Wall to extend along the length of the River to reach the coast.