Thanks for the advice and comments. I spent a couple of hours this
afternoon lifting a few tiles on the roof in the area where the leak
is to see what sort of state it was in underneath. There were a few
pools of water, the battens were wet, but no obvious place where the
water could be getting in.

The existing tiles are around 1" thick (and 12" square) and are
corrugated with interlocking grooves on either side (left/right). The
slope on the roof is only about 8.5 degrees from the horizontal, the
lowest-profile tiles I have found specify 10 degrees minimum
(www.decra.co.uk). As such I don't think retiling will definitely
solve all the problems. What I'm thinking about now is fibreglassing
the roof, and then replacing the tiles on top of the new roof. That
way I get to keep the look of the original tiled roof, but should be
completely watertight and durable.