Firstly, if you intend to lodge a complaint, I am sure you are dissatisfied and want an outcome from the complaint. With this type of cases whereby the workshop takes a long time to repair your car, I'm afraid this sort of complaint won't incur much damage to the workshop, if any. It's not like they are going to get fined by the insurance company or authorities, perhaps just a verbal or written warning at most I suspect.

The most you can do if you want to complain is to post a negative review of the workshop on Google Maps or somewhere else, and negative ratings will reflect on their incompetence. FWIW it's important to select good workshops that deal with accident repairs as some or most of these workshops have poor workmanship and customer service. If unlucky, your car will be stuck for a long time in the workshop. Good workshops will not only repair your car within a short period but the workmanship will be good also, and any minor flaws in the repair they will fix it for you without any hesitation. I have encountered some of these workshops where the boss himself is very friendly and helpful and the car was delivered within a short period, about 1 week only if I'm not mistaken and the damage can be said to be rather major.

To avoid such complications in the future, check reviews of the workshop that you intend to send your car for accident repairs on Google maps. Those bad workshops would have many negative reviews and it is hilarious to read some of the comments posted in those section by owners who have sent their cars there.

What is the name of the workshop? Just an example, here is a link to a workshop with poor reviews. You can post a negative review of the workshop which you had an unpleasant experience with here.

https://maps.app.goo.gl/981rTjWyt6Yo7WQp7