Gas burns more completely than wood and produces practically no PM2.5/soot. There is a little soot that builds up over time if the gas does not combust completely, but most gas fires are set to do this. The clue is in the flame, a blue flame is fully burnt methane, a yellow flame has more unburnt carbon. The reason gas flues don't need to be swept anywhere near as much as coal/wood flues is because they do not produce anywhere near as much soot.
I'd agree wood has less net CO2 impact, unless the methane source is bio methane (most of ours isn't).
I think the plan is to try to replace say 20% of the methane in natural gas with green hydrogen generated by renewables, but my guess is this is 10 years away at least. The great thing about it is that it will not really require any change in the equipment infrastructure. To go to full hydrogen will need new boilers.
The problem with wood/coal in cities is the PM2.5 they produce. Basically in the 50s we got killer "pea soup" fogs in London because of the sulphur dioxide from the coal and the PM2.5s just made that worse. The Clean Air Act reduced the coal burning to improve the air quality. Wood produces a lot of PM2.5s but does not produce any sulphur dioxide, so we won't be going back to the 50s, but the PM2.5s are a big problem. We've spent years trying to reduce the PM2.5s from traffic and diesel cars in London and been pretty successful. But now we are adding them back in to the atmosphere through wood burning, negating all the recent good that has been done in improving air quality. At some point in the future, there will be a much higher PM2.5 from wood burning, and one fateful night will come along with cold still air and the PM2.5 pollution will not be able to disperse properly, it will just linger in the streets like the pea soup fogs did. It's not hard to figure out what happens next.