For carbon emissions alone sure, but there's also the environmental impact of manufacture and disposal to consider, congestion and air pollution (tyres and brake pads), deaths and injuries on the road, concreting of green spaces...
Indeed, Highways England (responsible for 4,800 numbers of network) had the lowest level of emissions in the last 5 years of 330,000 tonnes CO2e (total of 2,100,000 over that 5 year period, so an average amount of about 400,000 tonnes); whilst Network Rail (responsible for 9,800 miles of network) had a CP6 target of reducing from a baseline of 300,000 tonnes CO2e, and achieved a 18% reduction (resulting in a sub 250,000 tonnes figure)
Likewise the maintenance and use emissions for a 60 year period of motorway of comparable length of motorway (using current emissions, which will fall in time) to HS2 phase 1 are about 22 million tonnes. If we could build 5* the amount of rail infrastructure as Phase 1 (5*1.5 million tonnes CO2e for building & 120 years of maintenance and use, so 7.5 million tonnes) but could then close a motorway of comparable length to just phase 1, that would reach 7.5 million tonnes of CO2e savings after about 20 years, even if road emissions halve it's still better after 40 years (however that would likely be an average of half, as it's unlikely to have halved by the time of opening so in the latter years of would have to be more than half.
However you've still got the remainder of the 120 years of rail use of emissions which are then effectively free.
For road to get close to the rail's figures (bearing in mind that we're building 5* the number of miles of new HS lines compared to the length of motorway which is being closed) emissions would have to be cut by at least 1/3.
However with that much extra track (which would likely need to improve capacity for freight to get to the WCML and passengers to get to HS2 services) chances are other road use across a much larger area would reduce.
For instance if we had a plan to create a rail system which allowed us to remove enough road traffic to close the M40 between London and Birmingham, that would likely need better freight connections (which would also benefit passengers) to HS1, Southampton and Felixstowe along with better passenger rail connections across Birmingham between Oxford and London/Birmingham and a reduction of some key pinch points across the whole network.
In reality you'd be hard pushed to actually close a motorway, however you could make it that there was enough of a drop in demand that some of the secondary road routes could be closed to through traffic and there be a lot more Low Traffic Neighborhoods introduced as carry use had fallen so much.
Whilst people may still use cars for occasional travel it could well be that these are shared use cars and so less road space was needed for parking (freeing up today space for moving vehicles).
Often things like HS2 get looked at in isolation, rather than how they could be used as part of a wider solution for the reduction in our car use. However that's probably because no one is thinking about that (hopefully yet) as part of our tour to net carbon zero target. As to achieve that we need to do as much as we can to reduce our emissions.
If we cut out emissions by 80% that's still 20% to deal with, however if we have scope to get it to 15% or even 10% then that's less which needs dealing with. Schemes like carbon capture are likely to be quite expensive so the more we can cut will help us.
That's going to likely mean that our focused should be (on order of priority):
- do we need to travel and reduce this where we can and it's practical
- walk as much of what's left as possible
- then cycle
- then use public transport
- then use shared car systems (car clubs, car hire, even taxis)
- then and only if none of the above are suitable (although may be harder for some trips, but still be suitable) then have your own car, however even in doing so for each trip you'd still need to try and reduce the use of it.
That's not going to be popular with a population who are addicted to car use and the level of convenience (and therefore the lack of planning required, such as being able to go to the shops for all their food shopping rather than being able to have a fortnightly food delivery and picking up half a dozen items once or twice between those major deliveries - whilst you may need milk and fruit and veg the big, such as toilet rolls, or the heavy, such as washing powder, should be delivered, as such the top up shopping could be carried whilst walking or cycling by most people).
HS2 (any phase) by providing more rail capacity and improving journey options by rail then allows more occasional travel to be undertaken by rail, as such it reduces the need for car ownership from some people (i.e. those that justify car ownership because going and visiting family by train is slow/difficult and so use it day to day when they don't really need to). The more we improve rail and other public transport the more we'd be able to reduce our need for cars.