The second F1 Esports Series will switch over to the still-in-development F1 2018, during the competition's team-based Pro Championships phase.
As stated in the Q&A section of the F1 Esports Series' website, F1 2017 will be retained for the four online qualification rounds. The wording in the Q&A also implies F1 2017 will be used for some of live Pro Championship rounds – with F1 2018 replacing the current game later in the year.
At time of writing, it's unknown when F1 2018 and the F1 Esports Series' Pro Championships will launch respectively. With the final qualification round closing on 6th June 2018, it's likely the Pro Championships will commence sometime in July, with the release of F1 2018 potentially following shortly afterwards.
The first of the four qualification rounds for the F1 Esports Series is currently running in F1 2017. Applications are also open for the upcoming online and offline F1 2018 closed beta tests here.