Is it possible to splice on to one of the hots of a base board heater for a plug?
You cannot and must not take a phase from one cable/circuit and a neutral from another.

Amongst other issues, this:


Poses a threat to any maintenance worker who disconnects the neutral or earth of the original circuit, assuming that because they have isolated the hot, the neutral is not carrying current.
Can overload the neutral, which does not directly have overcurrent protection.
Can cause issues and heating in surrounding steel as the magnetic fields do not cancel.
Will cause immediate tripping if either circuit is converted to GFCI in future.
Violates the NEC/CEC/any other sensible rules.