No, there is no timer for the next lesson. New lesson material is dependent on you getting a radical (blue background) or kanji (pink background) item into guru status.
It seems that you do not understand the process very well. Please take the time to read the following pages. To get the most out of WaniKani, you have to make the effort to learn how it works.
Unofficial FAQ - questions not found in the Official FAQ.
Regarding userscripts like “WaniKani Ultimate Timeline”, to learn how to install a userscript, please read this visual guide on how to install a userscript.