It was six months from Rome to Antioch, and about the same time from Antioch to the Rhine (referring to the back and forth campaigns of Severus Alexander I think). Rome to Hadrian's Wall was around 2/3 of that distance between Rome/Antioch, and the road systems would have been very well developed in Gaul and there was a much shorter sea trip involved. On the basis of that I would estimate the trip would take 2ish months, but I don't have a source for that specific claim. I would welcome estimates from others.

Source (for Rome to Antioch/Rhine travel time): Mattern, Susan P., Rome and the Enemy: Imperial Strategy in the Principate, University of California Press, Los Angeles, 1999.

If you're curious about travel times in the Roman world, check out the Orbis project from Stanford university.

It is an interactive map of the Roman world that allows you to simulate travel from any known place to another, using what we know of roads, river connections, shipping speed, etc. Very cool.

You can set what speed your imaginary soldier is moving at (does he have a horse? Is he making a swift military march?) what naval transport options are available to him, and what season he is travelling in, and you can select a route based on speed, cost or distance.

The estimates may sometimes be a bit optimistic (In real travel, things could and would go wrong, from blisters through dodgy roadside food to washed out roads to shipwrecks) and in this thread u/Hergrim notes that their average speed for horse travel seems rather high compared to historical guidebooks and military manuals.

Still, it's a lot of fun and very informative to use, and based on exhaustive research.

Incidentally, a quick test for your journey from a town near the middle of Hadrian's wall (Coriosopitum) to Rome gives a travel time of about 30 days in spring, most of which is spent sitting lazily on a boat. If your soldier has to march, it'll take more like 80 days. Still a lot less than the 6 month figure u/PapiriusCursor gave, which illustrates the difference between moving individuals and moving armies on campaign.