No public infrastructure will work for that - bus drivers will quickly die of radiation sickness or just refuse to do their work, but for one, one-way trip to save your own ass it's okay.
Essentially, this is why suburbanisation come to being: war planners realized exactly the thing you are pointing out: that evacuating people in a crisis - while the Soviet bombers will be flying (it was before ICBMs) was an impractical idea, it will kill more people than it will save; but what works instead, is to evacuate people in advance: just make them live far from city centers which can be attacked. And redline the city centers: let only people you don't care about surviving, live there.
In 1950s Soviet Union, "workers and peasants" lived in villages and suburbs, these were replaceable anyway and killing them had little benefit. Difficult to replace cadres - engineers, most qualified workers, high ranking military, Party officials and such - lived in posh Stalinist apartments in city centers, and would perish inevitably. In 1950s U.S., it was already other way around. Soviet Union could do nothing about that: they simply couldn't afford an individual house, and thus inevitably, individual car, for everyone - they still can't. It was a clear win and moved a balance in the Cold War a great lot in U.S. favor.
One thing to realize here, it will best of all work if U.S. attacked first - then they choose the timing and attack in late evening in the U.S. when everyone is at home already, are warned, and don't leave their houses in the morning to avoid being caught in road panic, and have some hours to prepare for the Commie attack such as letting go of any combustibles, covering car so it doesn't suck fallout particles in, and stocking water and if possible, food.
Hurricane evacuations even a week in advance are horrible by car. Highway infrastructure simply cannot cope with large movements of people in one direction.
Mass transit is cool, but evacuating an entire modern city on busses isnt viable. (Inter-city rail connections cannot be scaled up in such times.)
In the evacuation model those buses drive out a day full, and then need to drive a day to get back before they can pick up more people. It just doesn't work. No transit system has enough buses.
That is before we point out that if you have anywhere near that many people served you probably have some rail systems that cannot operate outside their lines (third rail power or overhead electric)
Cars have the advantage of there are enough of them. Note that in most evacuation situations the cars are full. (the roads don't have the capacity, but the cars do)
See the recent evac of fort mcmurray canada. That was all cars for hundreds of miles. No hope for busses there.
So the assumption that "90% of people won't need government help" clearly failed to hold true for this relatively small evacuation of 88k people.
The fact remains, most people who are told to evacuate by car will pack their suitcases and enough spending cash so that they can live on their own once they get out. The majority will end up sleeping on the floor of friends/relatives, so once you get them out of danger you don't think about them again. All that is required is enough warning that they can pack (if you don't have this much warning nothing can work anyway), and enough fuel (accounting for gridlock!) along the way.
Government needs to deal with the small number of people who cannot care for themselves, but the vast majority of people can be self sufficient, and having their personal car makes them more self sufficient.
More 25% is not a small number. A car only helps if people can actually get out. A mass evacuation by car of LA, San Francisco, Seattle or NY would fail utterly. Any place that already has significant traffic issues during rush hour (the sorts of places where buses are needed) would face massive drops in the evacuation rate due to gridlock and the areas that they would be evacuating through would rapidly run out of supplies (as happened with Fort McMurray with only 88k people being evacuated).
We shouldn't use bus when they are available for evacuation? Why? If we can get all the bus to be used + add bus from other cities nearby, why should we prevent people from using them? This is scaling much better and people with cars are free to leave with their own cars.
"Cars have the advantage of there are enough of them." --Surely not the case in many urban cities.
New Orleans had huge numbers of school buses. They went unused and were destroyed by flooding.
You might demand that the government not be incompetent, but the reality is that government will screw up badly. Individuals with cars can screw up, but the effects are limited to the people screwing up.
This idea have been marketed and for a good reason, it was good way to save U.S. population from Soviet nuclear threat - in a way Coomies could neither counter, nor replicate on their side - when Commies launch their nukes, it's much better if most people find themselves in separate houses with large tracts of land and little combustible materials, and good basements to protect from fallout, and in a few days, can emerge from there and drive their cars in whichever direction civil defense points as safer.