The Standard map should update within two minutes of uploading changes. You may need to force refresh (ctrl+shift+r) to clear any cached tiles. As for the other styles, these are generally updated less frequently as they're either hosted by other organizations. You can check who hosts 
which style by looking at the attribution box on the bottom right. The two bike oriented styles listed on the website are made by different organizations: CyclOSM is made by https://cyclosm.org and OpenCycleMap is made by https://thunderforest.com