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According to the S-Type 1999.25 Model Year Electrical Guide the Cooling Fan Activate Request from the Powertrain Control Module is Ground for Active and B+ for Inactive, not PWM.
I have checked the Coolant Temperature, which is taken from the Cylinder Head Temperature sensor, and it indicated 24 °C so that can't be the problem.
I had previously checked the A/C Pressure Sensor and that was OK.
This leaves the Engine Oil Temperature sensor to check.

 

Maybe the wire is chafed to ground? The fault may not necessarily be a bad sensor feeding the PCM. Could be a bad wire from any of the relevant sensors, and the PCM is responding as designed. Or there could be a short to ground between the PCM and fan module, with the fan module thinking this erroneous signal is commanded by the PCM.

 

Get a code reader (ELM-327 or similar) and read the Fan demand value out of the PCM. This will tell you whether the PCM is demanding the fan on or not. If it isn't, then you either have a chafed wire to ground or faulty PCM.

If it is demanding the fan on then it's either A/C pressure sensor or temp sensor. I'm not sure oil temp is used on 3.0 cars. You can read A/C pressure and temp out of the PCM also to see if that's the case.

Can you read the Fan Demand values out of the PCM?

 

In a few days I will be leaving for the Jaguar National Rally being held this year in the central NSW town of Bathurst, home of the famous Mt Panorama racing circuit.
I will be travelling South from Brisbane and meeting my brother-in-law who is travelling North from Melbourne. 1080km trip for me, about 800km for him.
He also has a 1999 Jaguar S-Type and will be bringing his iCarsoft LR V3.0 diagnostic tool with him.
In between events we will be diagnosing the continuously running fan issue.
I'll report back when I have some further news.