Cold conservatory - What to do

Not I, but relations have. It was a massive improvement over a polycarbonate roof. It's now usable as a functional room where it was only possible to use it on select days when it wasn't too sunny and wasn't too cold.

Exactly that.

OP, you have several types of heat loss happening in this relatively small space and unless all of the following are dealt with well then this space will always suffer.

1. Conduction heat loss: Because of the relative small volume to large external element area (effectively all 6 sides are heat loss elements) and a lot of these elements are glazing there is massive conductive heat loss happening.
2. Thermal bridging: Generally these conservatories are built on to an existing external wall there is a large linear thermal bridge at the junction of the conservatory wall/roof to the external wall.
3. Convection heat loss: Typically these glass roofed conservatories are not made air tight especially at the glass wall to roof junction, so again a lot of heat loss by air leakage.
4. Last but not at all least is radiation losses due to being surrounded by relatively very cold glass and our skin is very sensitive to this for of heat transfer. Even if you get the above 3 items under control you will never feel comfortable even if the thermostat says it's 21 degC in there. If there are blinds or curtains, use them to help with this. Ultimately though replacing the glazing with superior glazing (u-value ⇐ 0.8W/m2k) is best.

If it was me and I was in my forever home and I needed this space for daily use, I would seriously consider replacing the current structure completely and building a proper structure in its place.