The Scottish border lies along the River Tweed, and has done so for a very, very long time. Officially, this became the border in 1237, and while there have been a few conflicts over where, exactly, it is, it's stayed roughly the same due.

Wales, on the other hand, is a bit more complicated. Its borders partly reflect Roman settlement, with Roman forts on the mountains in the west of England representing an early Welsh border. However, after the Romans left and the Welsh kingdoms of Gwynedd and Powys came into power, the border shifted into something that is more recognisable today. Conflict between the Welsh kingdoms (Powys especially) and neighbouring English kingdoms (especially Mercia) led to the creation of Offa's Dyke, a big trench meant to provide protection from Welsh raids. This became the border in the late 8th century. While the modern border doesn't quite follow Offa's Dyke, it does follow more or less the same route. However, over the next several centuries the border between England and Wales fluctuated, with Welsh rulers sometimes taking land from the English and vice versa. The border was firmly established in 1535 with the Laws in Wales Act, which annexed Wales into England and established the border between them. This border was partly determined by tradition, but also by linguistic tendencies - places that mostly spoke Welsh became Wales, and places that mostly spoke English were England. There were a few exceptions and a few locations that would be ambiguous for some time afterwards - Welsh enclaves in England, for instance, would remain until the 19th century, and the status of Monmouthshire wouldn't be firmly determined until the 20th century - but largely, the border was set at that point.

Sorry to add on to this, but what role would Hadrian's Wall have played in the modern/historical borders?

According to Stephen Oppenheimer in his book, "The Origins of the British; a DNA Detective Story," the borders roughly correspond to a long standing genetic divide with the peoples inhabiting the "Celtic" countries being primarily descended from a group that had a western Mediterranean ice age refuge, while the English tend to be descended from a group with a Black Sea ice age refuge. The extensive DNA evidence he uses seems pretty convincing to me, though I am very far from being an expert. Interestingly, the much later invasions (Celtic, Roman, Anglo-Saxon, Viking, Norman and the like) that we hear so much about seem to have contributed very little to the genetic makeup of the islands, less than 5% according to Oppenheimer.

Edit: downvoted without explanation. This is a common occurrence whenever I bring up Oppenheimer's work, I believe due to the fact that he basically rides roughshod over so many cherished ideas about the people of the British Isles and their origins. Fortunately, the DNA evidence Oppenheimer uses is "hard" science and not really open to interpretation. Add to this body of evidence the fact that the archaeological evidence, incomplete though it is, seems to back him up entirely.