To keep my car on the road, I am now running without any petrol tank on
the
vehicle, awaiting delivery of the new pump. The instruction sheet that
came
with my conversion says it is crucial that I never let the car run out of
petrol. How important is this?

This will depend on how your conversion switches off the petrol.
If it does it by cutting the pump out, then it should not matter whether or
not there is any fuel in the tank, but if it cuts the injectors, and leaves
the pump running, then if you run it without petrol expect pump damage from
the pump running dry.

I get the impression that LPG offers no lubrication but assuming the
engine
has been built for operation with Lead free fuel I can't see there being a
problem. Hopefully, the new pump will be fitted within a week but I'm
limiting my trips to an absolute minimum

When running on LPG there is no petrol going into the engine anyway, so it
matters not whether or not if you have a fuel tank on or not.
As above, the only problem running with no petrol in is possible pump
damage, and you cannot damage the pump if it isnt there!