The "New Recession", which hasn't truly started, isn't because of the war. It's because of the tightening and rate hikes that the Central Banks currently need to do because of their BRRR-Money-Printing and government borrowing during COVID.
These things are not a simple and single cause and effect thing. They are more complicated than being summarized in one sentence.
The war is one piece of the chain that leads to the ongoing recession. For example rates hiked because energy prices keep rising and there is an energy shortage and at the same time there is all this printed money. The energy crisis is caused by the war while the money printing is partly the COVID result and partly bad policies. The combination of these two will cause very high inflation if they let the interest rates loose... so they choose lesser of two evils and increase interest rates at the cost of causing recession.