[modern] 88 XJ40 3.6 - No Start when Cold

Have you checked if there is a spark at the plugs when the engine is being
cranked ?

If the Crank Position Sensor does not provide a signal to the Engine
Control Unit, for whatever reason, then the engine will not be provided
with fuel or spark.

Assuming there is no spark, as a first and relatively easy check, look at
the CPS plug & socket connector. It may be that the cold conditions are
having some effect on the electrical connections, causing ‘separation’ /
open circuit.

These conditions may also be having some effect on the circuitry which
provides power to the fuel pump. Check to see if fuel is being delivered to
the cylinders.

If neither of these initial checks proves profitable, then you are into a
more comprehensive troubleshooting regime.

Kind Regards,

John Tarrant.