I looked through a ton of sources but all of them seem to come to the same conclusion that in Europe, gasoline (called petrol for the non-Americans) is not used in significant amounts. As WFH seems to only affect vehicle fuel consumption and not other fuel costs, this doesn't seem like it will do anything significant other than political posturing. It may also explain why the EU put restrictions on oil quite quickly (relatively, compared to natural gas). Since fuel prices are astronomical in Europe (compared to North America), it may already have already caused downward pressure on motor vehicle usage (compared to North America) so much of the population already uses greener options (public transportation, rail, carpooling, etc.).
North America is a completely different story. Fuel prices are low because two producers of oil (Canada and the US) contribute massive amounts of gasoline to their local markets. So even though motor vehicle usage is high, supply of fuel and lower taxes contribute to usage. However, this also means that they are generally insolated from outside pressure in regards to oil pricing. The US almost immediately sanctioned Russian oil (and tried to encourage the EU to do the same) after the Russian invasion of Ukraine because they are not dependent on Russian oil at all. In fact the US has tried to sanction Russian oil in the last few years by CAATSA (which of course did not fare well with European countries). That is to say that this idea will not affect the US (or Canada) in regards to dependence on Russian oil.
As for tackling climate change, gasoline usage in Europe is likely negligible given the above source. This is because the cities in Europe are older and designed for walking. In the United States (and Canada), most cities were built in the industrial revolution or later. Cities were designed for cars and have a sprawling neighborhoods and suburbs. This means that even if people stopped commuting to work, they would still have to drive to do any basic activity since walking is not viable unless you live in the center of the city (and even then, newly added amenities will be added in a sprawling fashion). Car ownership and usage is orders of magnitude higher than most countries (likely contributing to the disparity on the chart). While this means that there is a lot of room for improvement, this is not a viable solution in reality unless cities are redesigned, public transportation is widespread, and people can access public transportation from their homes without much effort. Unless that happens, this is not going to affect how much people drive cars even if everyone* works from home.
Lastly, the idea that everyone, or even a large portion of people can work from home is nonsense. A nurse cannot work from home if their job is to serve people in a hospital. A garbage collector cannot work from home if their job is to collect garbage from households. A factory worker cannot work from home if their job is to build cars. A stocker cannot work from home if their job is to stock shelves in a grocery store. Etc. These are the jobs that most people have. Not the chosen few who work in offices. Now, obviously most of the jobs that I mentioned above can just be automated, eliminating the commute of a human operative. But that would result in these people being jobless, which is politically unviable.