I can not find any documentation for this - is it possible?
The help for the fields you've found in the TortoiseHg config dialog (thg userconfig) is:
Issue Regex field:
Defines the regex to match when picking up issue numbers.
Issue Link field:
Defines the command to run when an issue number is recognized. You may include groups in issue.regex, and corresponding {n} tokens in issue.link (where n is a non-negative integer). {0} refers to the entire string matched by issue.regex, while {1} refers to the first group and so on. If no {n} tokensare found in issue.link, the entire matched string is appended instead.
In other words, if you configure them like
[tortoisehg]
issue.regex = [Ii]ssue(\d+)
issue.link = https://www.mercurial-scm.org/bts/issue{1}
then you will have a setting suitable for the Mercurial project itself: if a commit message contains the text "issueNNN" or "IssueNNN", then TortoiseHg will now make that a link to the Mercurial bug tracker for Issue NNN.
For Bitbucket's issue tracker you will want a link like
https://bitbucket.org/<user>/<repo>/issue/{1}/
and then capture the issue number in the regular expression. This works because Bitbucket is smart enough to ignore the rest of the URL after the issue number -- you can write whatever you want there, or write nothing as above.
Very simple functionality, but also quite useful when you often lookup bugs based on the commit messages.