SQLcmd error 'Failed to initialize sqlcmd library with error number -2147467259'
After hunting around a lot more, I found a few articles discussing security and granting permissions that sent me on a wild goose chase and ultimately made no difference to the outcome.

The script continued to work outside of sp_send_dbmail, but would not work with it.

I don't recall how, but I have subsequently discovered that stored procedures the indexing script allegedly uses had gone AWOL (ones created by the file CommandExecute.sql), even though the indexing script worked fine outside of sp_send_dbmail.

Once I created those stored procedures, everything seems to work as it should!