You could use a combo of the following, cooked into a script:
From How can I know in git if a branch has been already merged into master? You can use the git merge-base command to find the best common commit between the two branches. If that commit is the same as your branch head, then the branch has been completely merged.
Take that output (honoring above criteria) and do a git log -n 1 --pretty=format:%cd branchName to print the commit date
For example:
git branch --merged | grep otherBranch
If above is true (ie, otherBranch was merged), then
git merge-base refs/heads/master otherBranch | xargs git log -n 1 --pretty=format:%cd