If the matter is as you describe, and you cannot find anyone to take you to the mosque even in return for payment that you can afford, then this is an excuse that permits you not to attend the prayer in congregation, and there is no blame on you, in sha Allah.
But if you know that you have the skills in driving and are able to do that, and you are certain that no harm will result to you or anyone else from your driving on your own to the mosque, and the delay in obtaining your licence is only because of general administrative rules that require you to wait for the period mentioned, with no exceptions for anybody - if the situation is as we have described, then we do not think that there is anything wrong, in sha Allah, with you driving the car by yourself for this short distance so that you can attend the prayer in congregation.
But we still do not think that you have to do that.
And Allah knows best.
Reproduced from Islam QA