It is possible to resign at any point although you would be expected to serve your notice period first, so even if you resigned today you would be required to work for another week until your notice period expires. The issue then is that your employer could still commence disciplinary proceedings in that time and even dismiss you before the end of your notice period. Therefore, even if you had initially resigned, the actual reason for termination could end up as dismissal, if the employer completes the disciplinary process before the expiration of or notice.
Saying that, they would still be required to conduct a fair disciplinary process and that would include conducting a reasonable investigation, holding a disciplinary hearing and allowing you to defend yourself. They may not necessarily be able to do the whole thing in a week, although it is possible.
In any event, if they wanted to dismiss you they must show that a fair procedure as followed and that there was a fair reason for dismissal. As you have more than 2 years’ service you would be protected against unfair dismissal so to dismiss for gross misconduct the employer must show that your actions had been serious enough to justify that. Fitting the parts to a car which resulted in the car not starting is unlikely in itself to amount to gross misconduct. It would depend on what was actually wrong but this is not the same as for example fitting brakes that did not work and placing the customer in danger by having their car not being able to stop. You may have made an error in the installation process but if that is something which can be fixed and did not place the customer in danger or been such an obvious error that seriously affects the reputation of the garage, it should not be dealt as gross misconduct. Assuming you have a clean record to date this should really be dealt with through a written warning at most.
So you can hand in your resignation if you wanted to but bear in mind the requirement to work the week’s notice period. In the meantime if the employer decides to instigate disciplinary proceedings you can monitor the situation to ensure they are conducting a fair process and also defend any allegations with the factors I discussed above.
I hope this clarifies your position? If you could please quickly let me know that would be great, as it is important for us to keep track of customer satisfaction. Thank you