This is sort of a complicated question that gets to the heart of what we call "impoverished" and the way that underdevelopment is actually something that follows from development rather than existing prior to it. It can be said, briefly, that elite wealth in the northwest provinces (which is basically Britain, non-Mediterranean Gaul and Germania Inferior and Superior) was significantly less than what is seen in the Mediterranean provinces. Which is not to say there weren't some quite wealthy people, as these areas do have some enormous villas, but nothing like what you would find in Italy, Asia Minor or the other Mediterranean areas. There were, in general, lesser signs of material prosperity, both in rural areas, sen by the size of villas, and in the cities, which tended to be much smaller and less richly adorned. By and large, the farther you get from the Mediterranean the farther outside of the cultural and economic mainstream you become.

So in this respect, yes, these areas were underdeveloped and backwards. But when examining osteological remains, one of the best ways we have to understand health patterns historically, the average in Britain is higher than in Italy (I think the sites are from Western England and central Italy, technically). People tended to be taller, live linger and be less afflicted by disease. That is probably because these areas were less overpopulated, less subject to warm climate diseases, and more free of extreme wealth and power hierarchies.

So while these areas were backwaters, they also seem to have had higher average standards of living.

I'd say that northern Europe actually became richer after the fall of the Roman Empire - but this was due to inventions and land usage rather than the fall of the Roman Empire itself.

The invention of the heavy plough and the horse collar/harness allowed deeper plowing that increased agricultural production in northern Europe quite a bit. Being able to sow deeper and in heavier (fertile, but otherwise hard to work) soil in a way that was more protected from early spring and late autumn fost was quite an advantage.

At the same time, over-grazing and deforestation in the Meditteranean area created vast tracts of macchia land - eroded soil covered by low, thorny brushes that grazing animals are reluctant to eat. This reduced agricultural production around the Meditteranean.

This meant that during the early medieval period after the fall of the Roman Empire, the north started to become richer rather than poorer, especially relatively.