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Yes, there can be a delay. You need to follow the suggested steps first.
The cape typically appears in Bedrock Edition first. Here are the steps for checking it.
Open Minecraft: Bedrock Edition and check Dressing Room to confirm the Aurora cape is present there.
If it is present, close Bedrock and open Minecraft: Java Edition.
Go to Skins.
Highlight the current skin.
Select … > Edit.
Choose the cape and Save & Use.
Restart Java Edition and check again.
If the you've completed all the steps but the cape still remains unavailable, contact Minecraft Support and request that they check the cape entitlement on your account. Include a screenshot of the completed redemption and the approximate date you claimed it so they can investigate further.