For me this messed up all the commits before branching from the main branch. After force pushing GitHub didn't recognise the main and feature branch's common history. The message was:

There isn't anything to compare.

Instead of this I had to rebase with keeping the original commit times, then force push.

git rebase --committer-date-is-author-date main feature
git push -f

Only after these steps could I create the pull request.

To anyone using the initial answer here, I would also suggest that you figure how many commits you actually need to and should rewrite. I made the mistake of running this on a long commit history, when I only needed to fix the dates of the last few commits.

The command that worked for me is:

git filter-branch --env-filter 'export GIT_COMMITTER_DATE="$GIT_AUTHOR_DATE"' {SHA of commit before first commit}..HEAD

Expanding on @juliamcneill's edit, if you also want to make sure the update commits are signed, you can add a --commit-filter to the command as well:

git filter-branch --env-filter 'export GIT_COMMITTER_DATE="$GIT_AUTHOR_DATE"' --commit-filter 'git commit-tree -S "$@"' {SHA of commit before first commit}..HEAD

If you're running into issues with a "Cannot create a new backup" command, modify it with -f before the filters:

git filter-branch -f --env-filter 'export GIT_COMMITTER_DATE="$GIT_AUTHOR_DATE"' --commit-filter 'git commit-tree -S "$@"' {SHA of commit before first commit}..HEAD

In case someone is feeling a bit sceptical using filter-branch, here are the alternate commands using filter-repo.

1. If you want to rewrite CommitDates of all the commits:

# Using filter-branch
git filter-branch --env-filter '
  GIT_COMMITTER_DATE=$GIT_AUTHOR_DATE
'

# Using filter-repo
git filter-repo --commit-callback '
  commit.committer_date = commit.author_date
'

2. If you want to rewrite CommitDate of a specific commit:

# Using filter-branch
git filter-branch --env-filter '
  if test "$GIT_COMMIT" = "<SHA_of_commit>"
  then
    GIT_COMMITTER_DATE=$GIT_AUTHOR_DATE
  fi
'

# Using filter-repo
git filter-repo --commit-callback '
  if commit.original_id == b"<SHA_of_commit>":
    commit.committer_date = commit.author_date
'

3. If you want to rewrite CommitDates of a range of commits:

# Using filter-branch
git filter-branch --env-filter '
  GIT_COMMITTER_DATE=$GIT_AUTHOR_DATE
' <SHA_of_commit_made_before_the_one_you_want_to_rewrite>..master

# or three commits from the top
git filter-branch --env-filter '
  GIT_COMMITTER_DATE=$GIT_AUTHOR_DATE
' master~3..master

# Using filter-repo
git filter-repo \
  --refs <SHA_of_commit_made_before_the_one_you_want_to_rewrite>..master \
  --commit-callback '
    commit.committer_date = commit.author_date
  '
# or three commits from the top
git filter-repo \
  --refs master~3..master \
  --commit-callback '
    commit.committer_date = commit.author_date
  '

You can read more details here.