After I purchased the item, the seller neither shipped it nor replied to my messages. This went on for about half a month, so I filed a refund request with eBay. Even after I requested cancellation, the seller still didn’t reply to me. Could you tell me roughly how long it will take for me to get the refund?

Did you ask for a cancellation or did you open a case using the Ebay Money Back Guarantee to say you had not received your item? These are different processes and neither automatically refunds you. If you open an 'item not received' case and the seller doesn't prove delivery or refund you, then you must go back to that open case and ask Ebay to step in.

Request for Help Regarding Refund | Member To Member Support

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Request for Help Regarding Refund | Member To Member Support | eBay UK