Not for heating oil, since most installations that use gas for heating do not gace the capacity to switch to oil. And those who have the capacity to switch their heating system would choose a heat pump over oil (better yield).

But oil (&coal) may be bought to generate electricity (to replace gas, not for heating but for electricity production. Some European countries such as Germany are heavily reliant on gas for electricity production), and that may still drive petrol up

I suspect most houses don't have the capability to switch between different modes of heating: if you have a gas furnace, you're not going to be burning heating oil in it instead.

I believe the plan in case of shortage is to reduce the amount of natural gas flowing to power plants, and to make do with whatever alternatives for electricity are readily available (which is probably coal, although maybe nuclear decommissioning will be staved off).

No, because the types of fossil fuels do not substitute well. You can't use oil for heating in Gas burners, industrial processes etc.