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I'd be grateful if someone could advise me how to find where the above lady is buried. I know that she died in a nursing home in Dundonald, Scotland in 2002. Unfortunately, the nursing home records no longer go back to 2002.
I am doing this research on behalf of the ex-husband of the above lady, who only recently found out that she had passed away. I've searched Ancestry for the burial record, but without success. I had thought of ringing local funeral directors in Dundonald, but thought that they may not have retained their records either or may be restricted from divulging the information for a relatively recent burial.
Any advice would be appreciated. Thanks in advance.

You could try local newspapers for notice of the death which may also carry funeral details.

Also try speaking to local funeral directors who may be able to give you the information. You might be surprised as to how long they keep records. They may be able to pass on a letter from you to whoever arranged the funeral.

Also try local Cemeteries/Crematorium.

Likewise the death certificate may name a relative who you could trace. You would have to order that from the Records Office.

Thank you greyghost for your suggestions. I'll certainly follow up on the local newspapers and the cemeteries/crematoria