Given that electricity costs about 4 times a kWh of gas, it’s probably for the best that it has a COP around 3. It will be more when the air outside is 21C. The economic argument for ASHP relies heavily on the excellent efficiency when it’s relatively warm outside. It’s not so hot, figuratively speaking, when it isn’t.
In my experience with air source heat pumps in the U.K. is where the air temp is between 2 and 8 degrees and high humidity. This this point a lot of ice forms on the outdoor coil requiring regular defrosting (where the compressor generates heat in the outdoor coil to melt the ice). At this point it’s backup electric heaters (as you say, COP 1) plus the power for the compressor. Once it drops below 0C it improves again as humidity drops off as the water is already ice on other surfaces like cars.
Just like PV and solar hot water, ASHP has limitations due to the British climate, but overall, it is more efficient, but not by as much as the headline 300% suggests.
In carbon terms, it really depends on how the electricity is generated. On average a kWh of electricity generates about 0.233 kg CO2e where as 1 kWh of gas around 0.184 kg CO2e. So slightly better in those terms too.