Should we wait until the tool is ready before we start delivery?
No. Don't wait. Start delivery as soon as possible.
Use this as an opportunity to get your process right. Ask learners now how it feels to complete the mental health tools (PHQ9, GAD7, WEMWBS) and learn from their responses. Remember to ensure that there is someone in your team to help people who need help to complete them but remember these are self-assessment tools for learners. Your 'administration' is making sure they are available, giving them out, collecting them back in, discussing them, thanking people for them and keeping their scores stored safely for transfer to the evaluation team.
When the learner survey is distributed, you will also need to get each learner to complete that at the beginning and end of their course.