I have an old B&D cordless drill; the batteries don't charge.

Either the batteries or the charger are faulty.

I've opened up the charger. At the input of the transformer, I measure
240V. At the output, 0V (I'd expect 18V or so).

I assume this means that the transformer itself is broken - at least, I
can't think that anything in the circuit after the transformer could be
responsible.

I haven't desoldered the transformer from the circuit board, but I guess
that would be a way of being sure.

Is that correct?

Daniele