This week I wrote a lesson about using "Firstly, Secondly, Finally", and a student asked me whether it's ok to use "at last" instead of "finally".
The answer is no. "Finally" and "at last" are not used in the same way.
Use "finally" in the IELTS writing test when you want to make your final point, or to talk about the final stage in a process e.g. Finally, as well as making life more fun and interesting, new experiences can be good for our physical and mental health. (taken from this lesson)
Only use "at last" when you have been waiting for a long time for something to happen e.g. At last I've passed the IELTS exam!