Getting started with wavelet-based real-time anomaly detection for temporal signals using Python
I suggest that you look at the Wikipedia article:

https://en.wikipedia.org/wiki/Change_detection

and first focus on the statistical aspects of the problem.  The references cited reflect most of the foundational material of the topic.  The article is of high quality. You should also think about the systems aspect of your problem like, the architecture of the system in the sense of issues like having a set of essentially special purpose detectors, or something that allows some interactions.  Your problem statement says that only a single anomaly is present at a time, and this should be reflected in the organization of system.  Questions such as Do you need automatic gain control (AGC) or do you need a constant false alarm (CFAR) behavior do have  indirect effects.  

The major departure from the usual change detection problem, is $a(t)$. Typically, it is in $a(t)$ that one wishes to find changes.  I don't think that this is necessarily a problem but $a(t)$ will propagate through your detectors. 

Change detectors work on changes from something typical and $a(t)$ seems like a good place to look for "typical". This is probably where your likelihood ratio test or score metric is going to reside.

I'm guessing that your essential problem is that $a(t)$ is from a remote sensor, and your anomalies are associated with a transmission channel.  If so, transmission glitches are not typically subtle low SNR events that necessitate near optimal techniques needed to find them. They are obvious and are often managed by simple heuristics. Much that follows will be irrelevant if this is the case.

Finally, Wavelets. The answer is, you are going to determine this for yourself, by trial and error and thinking through what makes sense. One heuristic is that time-bandwith matching has a small loss relative to something like a matched filter.

The choice of basis functions is specific to your application.  People use Wavelets. People use STFT, People use Wigner.  People use Cyclostationarity. People use running histograms.

It all depends on how parsimonious a basis set represents your signals of interest, but you can't neglect $a(t)$. Those basis function have a direct bearing on characterizing "typical", which is why I strongly recommend you think about the statistical organization of your system, and then work on selecting your basis functions.