E-mailing the Senior Lecturer?

I think this is something that will depend completely on the circumstances. Given that you probably shouldn't share that here, you should probably ask your tutor whether they think it is a good idea.

A few words of caution. Probably few if any are in a position to predict how cases will be handled by the Senior Lecturer and her office. They take decisions in thousands of cases each summer, and the time taken to process cases varies considerably.

You should not assume that you have nothing to lose by emailing the Senior Lecturer yourself, as well as making the case through your tutor. It is probably fair to say that a reasonably prompt decision is more likely in a case that seems straightforward and uncomplicated on the basis of the paperwork, or that seems both clear-cut and urgent. A case file containing a separate plea from a student arguing a case "fully and comprehensively" is less likely to be viewed as the sort of case that is best dealt with through a summary decision by the Senior Lecturer, and more likely to be viewed as the sort of case that would be more appropriately dealt with through a Court of Appeal. And Senior Lecturers in the past have in more involved cases issued rulings to the effect that the Court of Examiners should make a decision on purely academic grounds, in the expectation that an appeal will then be brought, resulting in the case made by the student though the tutor being considered by Court of Appeal whose membership will evaluate the case made by the student. Such cases only come to the Senior Lecturer for final approval if and when Courts of Appeal have first decided the case in favour of the student.

I've had several decisions about me made by the senior lecturer always through an application by my tutor. I'd stick with the tutor application alone. They know what they are doing. Also if your school supports your application more generally Senior Lecturer seems to seriously take that into account.

Once the senior lecturer gets involved all the cases she reviews have serious implications for the student typically, as you mentioned, whether or not continuing in college is permitted. Therefore, with respect, I don't think anything you have to say in your own personal application will be anything they haven't heard before. Make sure that your tutor, however, has made a full application which you are happy represents your case.

As to the length of time it takes to receive the decision I completely understand you might be getting anxious. To be honest the decisions can take weeks. I previously waited a month.