Add config for signing base-browser nightly in tools/signing/nightly

After #40763 (closed), we should add config for signing base-browser nightly.

added Doing label

assigned to @boklm

mentioned in issue #40762 (closed)

marked this issue as related to #40762 (closed)

mentioned in merge request !679 (merged)

marked this issue as related to #40758 (closed)

closed with merge request !679 (merged)

added Build System Signing labels