Just installed a new thermistor...

...and apparently my hotend temperature is approximately the same as the surface of the sun. The bed temp is at the correct 20C though. I haven't made any config changes. What might cause this?

After some Googling, I noticed a relationship between temperature and resistance... as resistance decreases, temperature increases. So if the temperature appears to be infinitely high, the resistance is probably really really close to zero - like, when the thermistor wire is shorted out. Low and behold, the screw holding the thermistor in the hotend was tight enough to expose the copper and short it.

Temperature is a brisk 23C now

There are much more values which influence the temperature readings. For me it was hard to get the original thermistor for the heated bed so I changed the config.g to fit my thermistor.
Without any details which thermistor you used it is impossible to help you. There are four values at total whose are influence the resistance at a given temperature. However only two are important. The other ones are for finetuning and I am not sure if you can set them at the firmware.

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Slicer: Simplify3D 4.0; sometimes CraftWare 1.14 or Cura 2.7
Delta with Duet-WiFi, FW: 1.20.1RC2; mini-sensor board by dc42 for auto-leveling
Ormerod common modifications: Mini-sensor board by dc42, aluminum X-arm, 0.4 mm nozzle E3D like, 2nd fan, Z stepper nut M5 x 15, Herringbone gears, Z-axis bearing at top, spring loaded extruder with pneumatic fitting, Y belt axis tensioner
Ormerod 2: FW: 1.19-dc42 on Duet-WiFi. own build, modifications: GT2-belts, silicone heat-bed, different motors and so on. Printed parts: bed support, (PSU holder) and Y-feet.
Ormerod 1: FW: 1.15c-dc42 on 1k Duet-Board. Modifications: Aluminium bed-support, (nearly) all parts reprinted in PLA/ ABS, and so on.

Not sure what the fourth one is, but I know of three:

R: this changes the resistor value on the Duet board itself. RRP advised changing this to 4700 for newer Duet boards
T: used for changing the thermistor value. I use this on my heated bed, since it came with a 100K
B: beta parameter. I'm not sure when this would actually be used, but there's an equation that defines the relationship between resistor, thermistor, and β.

Every thermistor has a specific resistance at a given temperature as well as alpha, beta and gamma values. Like I said before I wasn't talking about the firmware settings. I talked about the attributes of the thermistor itself.
To help we need to know what thermistor you bought.

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Slicer: Simplify3D 4.0; sometimes CraftWare 1.14 or Cura 2.7
Delta with Duet-WiFi, FW: 1.20.1RC2; mini-sensor board by dc42 for auto-leveling
Ormerod common modifications: Mini-sensor board by dc42, aluminum X-arm, 0.4 mm nozzle E3D like, 2nd fan, Z stepper nut M5 x 15, Herringbone gears, Z-axis bearing at top, spring loaded extruder with pneumatic fitting, Y belt axis tensioner
Ormerod 2: FW: 1.19-dc42 on Duet-WiFi. own build, modifications: GT2-belts, silicone heat-bed, different motors and so on. Printed parts: bed support, (PSU holder) and Y-feet.
Ormerod 1: FW: 1.15c-dc42 on 1k Duet-Board. Modifications: Aluminium bed-support, (nearly) all parts reprinted in PLA/ ABS, and so on.

Not sure, it came with my spare J-head!

I think I'm all sorted now, the thermistor is now showing the same temp as the bed at room temperature. The extremely high temperature was caused by a short in the thermistor wire.

Then normally the B factor is about 3950. You used this thermistor for the heated bed?

---

Slicer: Simplify3D 4.0; sometimes CraftWare 1.14 or Cura 2.7
Delta with Duet-WiFi, FW: 1.20.1RC2; mini-sensor board by dc42 for auto-leveling
Ormerod common modifications: Mini-sensor board by dc42, aluminum X-arm, 0.4 mm nozzle E3D like, 2nd fan, Z stepper nut M5 x 15, Herringbone gears, Z-axis bearing at top, spring loaded extruder with pneumatic fitting, Y belt axis tensioner
Ormerod 2: FW: 1.19-dc42 on Duet-WiFi. own build, modifications: GT2-belts, silicone heat-bed, different motors and so on. Printed parts: bed support, (PSU holder) and Y-feet.
Ormerod 1: FW: 1.15c-dc42 on 1k Duet-Board. Modifications: Aluminium bed-support, (nearly) all parts reprinted in PLA/ ABS, and so on.