my first thoughts are either, sele oil level, pump relay, ot the pump itself
Oil level is easy to check, and I thing the pump would run if low on oil, just not get to enough pressure to change gear.
The relay is tucked under the large air hose that feeds the engine and between the barrery and thermostat, giving that a waggle can help, or swap with the IIRC rad fan relay to test it.
Finally the pump may have gone, which does happen on the older cars, the brushes to the electric motor simply wear down and no longer make good contact. A new pump in the UK is £300, straightforward job to do. Just have the pump tested (easy, run some wires to the battery and see if it makes a noise) before going to the expense.