For this, please check Bedrock edition first and if it is there then you can try equipping the cape in the dressing room and then reload the launcher/website then check it on Java.
Also, please take note that there might be delays specifically when there are other players worldwide which is doing the same thing and if it still has not arrived then you can also reach out to the Minecraft team at https://aka.ms/Minecraft-Support/ for your account to be checked as well.