Oil temp sender would be my guess, so it would be to do with the gauge yes 
Shouldn't do. It's more likely to be the wiring at the gauge end if at all.
I'd get it working if you can. It's nice to know your oil temp to stop you pushing the engine before its up to temp. Yours are nice gauges also so would be a shame to leave it broken.
You can get a new senser off ebay , looks like you have a oil cooler should be a take off point for
Your senser would be a better idea than the sump
Your senser would be a better idea than the sump
Water and oil
^^ The 2 most common temp sensors....
Yup, it's only two wires, easy to repair. Where's the wires snapped off?
Its broken at the sump plug inside the metal coil around the wire...I realised after I undid the 17mm plug that perhaps I should have undone the smaller 14mm nut of the sensor. If I remove the sensor leaving the 17mm plug still screwed in will the oil come out or is the sensor sitting inside the bit of the plug that sticks out inside the sump?
The oil will still come out. The 17mm has a hole down the middle for the sensor to go down. The coil is just a spring, you should be able to slide it up, revealing the cable, if it has any cable hanging out you can repair it, if it's flush, youll need a new one.
You have to plug out the harness for the senser, mine broke in the same place cant be joined.
For what it costs you might as well drain and replace the oil and replace the sensor. Fairly easy job so shouldn't cost a fortune even if a garage do it for you.