No, there will be a long timeline, because the first "self driving" cars will be level 3, and for a robotaxi you need a very solid level 5.
Do you though? Truly?
A Level 5 system needs to be able to handle practically any scenario, but it probably isn't needed for everything. Take these 2 scenarios.
Scenario A:
Downtown Manhattan. I need to call a cab and have it pick me up at the corner of 3rd and 13th and then drop me off in front of Penn Station.
Scenario B:
Downtown Manhattan. I need to call a cab and have it pick me up at the corner of 3rd and 13th and then drive me to upstate New York and drop me off in front of a house on a private dirt road.
In Scenario A, a L4 system could probably be made to handle that just fine, hell even a L3 might be able to do it without much problem though it might be sketchy at points. You could probably doze off to sleep in a L4 in this scenario no problem.
In Scenario B, a L3 system could probably handle 90% of this but you'd probably not want to sleep. A L4 system would probably be able to handle all of this up till the private dirt road and then you might have to either give more guidance or take over.
The main point here is that we could have robotaxis with no drivers rolling around way before L5, it'll really just come down to knowing whether the current level of the system can handle the zone its operating in just fine. You only need L5 when you are trying to do more extreme stuff.