Further update, which might be useful to others.

Even running with a variac the drill would cut out after some minutes
work. This was fairly consistent and originally I attributed it to the
drill getting warm, guessed that there was a thermal cutout operating
somewhere. I'm cutting a 1mm or so piece of steel so I did think I
might be overloading the motor. After half an hour's rest or so the
motor would start up OK again.

Anyway, after a few of these cycles it seemed to pack up completely ;-
(. I buzzed through the motor windings and one of the stators measured
as open circuit. I was just about to throw it in the bin when I
thought to check the connection between the stator wire and the pin
into which the volts go. This is not soldered as I expected, but is a
sort of cheap IDC knife-edge connection, for the enamelled wire. The
connector fell off the wire as soon as I looked at it.

So I've remade this connection and things seem fine again. It looks
like when manufactured the connector is forced down into its housing
where it supposed to engage with the wire and 'cut' a connection. I
did this a bit more thoroughly than they seemed to have done at the
factory. If it fails again I'll probably solder it up.

I'm guessing that originally the drill warming up was causing this
connection to fail after a while, and be re-made when the drill cooled
down. Not very impressed with Dremel quality at the moment....

jon N