I know nothing about Series 1s, but I know a man who does. I emailed John Merton in Australia, and here's his reply:
G'day Tim,
Without seeing it, I'm at a bit of a loss. But...
The (new) common cable reference could be the clue. At the bottom of the handbrake lever there's a block which is slotted lengthwise for the cable to pass through. The slot is threaded - five sixteenth BSF from memory - and in front of this, drilled out to slide along the cable (a few may also be slotted) is another bolt which serves both to locate the cable in the block on the handbrake lever and as a means of adjustment. In front of this bolt is a stopper on the cable. When the handbrake is pulled on, the end of the bolt forces this stopper, hence the cable, forward, activating the rear brakes.
It sounds as if this has been incorrectly assembled. The order, from the front of the car back is, first the stopper on the cable, next the (head of) the bolt and then the block on the bottom of the handbrake lever.
Aside from this I haven't any real suggestions other than seeing the car, as it seems unlikely that the lever itself would have been disassembled when fitting the new cable. If it was, check that the return spring is working and also the orientation of the engagement pawl at the bottom, but it's hard to think of the issue arising here.
