Hey arik_17,
No worries, the Wurst Client is not detectable on Realms. Realms are like vanilla Minecraft servers, they don't have any plugins or built-in anti-cheat measures. So, you're good to go there.
However, it's worth noting that Mojang monitors the chat on Realms pretty closely and has been known to ban people if they see a chat message they don't like, even for minor offenses like profanity. I'd recommend keeping the NoChatReports feature enabled on Realms and not saying anything in chat while you're there.
Hope that helps, and have fun on your Realm!