The trouble is that the electricity demand is always topped-up by gas-fired generators. It's always there at the margin. Things like nuclear and renewables usually run at 100% of their ability, their generation is not affected by demand.

Therefore any increase in electrical demand is made up by turning up the gas-fired generators an equal amount.

So if a heat pump has a COP of 3, and the additional electricity consumed is generated from gas with, say, 33% efficiency (from the gas burner to the house electrical meter), there would be zero net change in overall gas consumption by converting from a gas boiler.

This is especially the case in winter, which is when most heating demand is, and the gas generators are definitely going to be running and making up that marginal generation.