Some of the series in the webtoon extension in Tachiyomi got limited to the first three chapters, will downloading the app unlock them all or will I have to wait weekly
No i don't think it will The extension for paid stuff is like that Like in bilibili extension for example, all locked chapter won't be shown at tachiyomi list even if you can access it for free in bilibili apps
If you mean Webtoon as in webtoons(dot)com aps/website, then nope, but if you buy the coins you'll be able to (I think/not mistaken) up to next 5 ch