ICE cars will go first. Most personal cars are only driven a small mileage per day and don't need to carry much weight. Needing to go more than 350 miles in a day, or needing to tow a trailer are statistical outliers to most users. Anyone who occasionally needs to go really far or carry more weight will find another solution.
As you said, I doubt ICE cars will go away completely. There's too much hobby/enthusiasm behind them for that to happen.
Trucks, buses, and planes will take much, much longer to phase out because of their extreme needs; e.g. light, compact power with a huge range.
Any other problems outside the vehicles themselves will be solved very quickly as they gain more market share.