Not quite sure what registration info you've actually got for the bike - is it just an old reg. plate with 101NE on it?
On it's own that wouldn't be enough to get the machine registered with the number, I think you'd need some documentation showing the frame number and the reg no. together - or the machine details would need to be on the DVLA system and if they were I don't think they'd have re-issued the number to a BMW. I suspect your Arrow was already off the road at the time it's details would have been needed to be put into the DVLA computerisation.
I'm no expert on the old registration number allocations - some people seem to be able to work out the date windows to within a few months as well as the local areas - but I think NE with a preceding number might have been issued by Manchester in the 1960s. You might have a chance of finding the original details if the local archives have included the motor cycle new registration records for the period from the old local authority sources - if they do exist it would be very interesting to see what the DVLA make of having two vehicle identities on the same registration number in these circumstances!
The only place I've found with a record of what archives are where is here:http://kitheadtrust.org.uk/england-wales
If you can get the dispatch details by asking for machine dating details (AOMCC, Drags or VMCC can supply those I think) you'd be able to confirm where the machine was originally supplied to.
There was a similar case on the forum a while back with a 'barn find' Leader that had a registration plate and I think also tax discs but no actual registration docs, that ended up having to get an age-related registration (on a different number) and you might well have to do the same.
'55 Huntmaster, '56 VH, ' 51 VH, '80 R100RT, '00 Sprint ST (but all those Ariel parts can only make one running bike...)