The latest roadmap from TSMC now points to 2023 for 3nm. This actually put Intel in the same time frame as TSMC. So incidentally both Intel and TSMC will be on 4nm in 2022, and 3nm in 2023.
Note: Intel 4nm is a Full, new Node while TSMC 4nm is an optimisation of their 5nm.
Of course the roadmap and time table from Intel is often not a reliable indicator and also doesn't tell you whether it is product shipment or if they are only ramping. My guess is that Intel is running as fast as they can to reach those target. That is also not including capacity difference with node from both companies.
I think Intel under Pat Gelsinger is making the same transition as Andy Grove did to during his era.