Hi,
It is still possible to work for a construction company as a subcontractor when you are waiting for your UTR number, as you will be deducted at the higher rate (30%) until the contractor is able to verify you.
Once your subcontractor registration is completed, any unmatched verifications will need to be matched to your record.
I would also refer you to Check when you can expect a reply from HMRC via the following link, -
Check when you can expect a reply from HMRC
Thank you.