Wouldn't it be useful to differentiate between culture and economy. While
the former would linger, the latter would be very much dependent upon links
to imperial administration, especially the demands of the Roman military. In
the 4th century, Britain acted as a supplier of grain, an offshoot of which
would have been a significant flow of cash into the villa system. However,
once Britain was removed from the Imperial system and was removed from the
list of suppliers, this flow would have ceased. Hence, the date when Britain
was no longer considered part of the Empire (perhaps before if supply routes
were cut off), would also mark a point at which singificant flows of money
stopped coming into the country.
Whilst Roman culture would certainly continue after this point, the ability
of the elite (especially) to fund that culture would have declined
significantly. And yes, certainly existing money supply can continue, but
without additional minting, the quantity of physical money will decline and
the value of individual coins will increase. At some point in time, coins
become too valuable to use on an everyday basis and they are instead ...
After that point is reached and likely before, significant trade once more
becomes a system of barter because you don't want to pass over all your
valuable coinage. So in order to get your Byzantian oil and wears, you need
a commodity to trade. Like tin. Or gold. But not wood. And not grain.
Lots follow I suggest.