I like Michael Wood, but any notion that the current Euro crisis has any relation to the fall of the Roman Empire is just silly. Trying to link the 2 will only serve to confuse both issues. I suspect that the BBC got Wood to write such an article to give publicity to a forthcoming program.
The immediate reason for the withdrawal of Roman troops from Britain was because a general wanted to use them to back his claim as emperor. More generally, the later Roman Empire seemingly lost the ability to raise, and pay for, soldiers, thus it could not stop barbarians invading. There seems to have been a decline in the economy and, I believe, quite a large fall in population. At the same time, the most wealthy people seem to have actually increased their wealth.
The Euro crisis, I think, is about individual governments within the Euro currency failing to be fiscally responsible, thus causing the Euro to be less valued.