Looking for some advice on whether we are allowed to use a safety harness that has passed its inspection date but is booked to have it again this week?
Pre-use inspection has identified that the Contractor should have been in last week to inspect them but when we checked the date the inspection was due later this week.

Going out to buy a new one now.

Is there any lee-way?

This is not complicated, most people who wear harnesses (myself included) would normally check before wearing anyhow. You look for fraying of the material, wear marks, is the stitching still in line, is the lanyard still operational, does/do the clip-on(s) still function, any bending in the metal of the clip-on, CE marking and product information tag still present, is the harness box still in use (does the harness get hung up to dry at the end of shift?), do you have harnesses with foot/leg loops (to prevent suspension trauma?) are they still intact?

It's not that complicated.

The people wearing the harnesses should of course have 'H' (for Harnesses) on their IPAF or other trade cards. If not ask them, have they ever had any training on how to wear a harness, if not, address it.