- Once can function as an adverb, adjective, conjunction or noun
- Since yes, can also function as adverb. Since can also function as a conjunction or preposition
- After can function as a preposition, adverb, adjective noun and conjunction
You are correct in the overlap between once and since, they can both function as adverbs and conjunctions.
- [Once I made] a cake from meat or [I once made] a cake from meat - once is the adverb describing the verb 'to make'. The description is that it was made in the past tense, hence the past participle usage of 'to make'. Once describes a single isolated instance in the past.
- [Since I made] a cake from meat I felt ill or I made a cake from meat and have been feeling ill [since I made it] - since is the adverb describing the verb 'to make'. Since can link the past to the present, implying that you still feel ill
Note that since is the adverb of the verb "to make", but in the sentence "I made a cake from meat and have been feeling ill since I made the cake from meat" there are a lot of unnecessary duplicate words. This can be shortened to "ever since", "ever since then", "since then", "since that time". However, the word still functions as an adverb of made, linking that time with now. The phrase "I since made a cake" is not used because there is no way to link then and now. You need to say "since the time I made a cake from meat" - "I felt ill".
- Call me [once] you get there - same as above, linking the phrases in a finite sense.
- He has been calling me [since] he got there - same as above, linking the phrases in a continuous sense. He has called me and he is still trying calling me.
This indicates a spatial, temporal, or other relationship. Since and After can both be used to indicate a temporal location.
- Since that time, since yesterday, since breakfast
- After that time, after tomorrow, after lunch
Again, since implies a past event that is still continuing and as such is often used with the Past, present and future perfect continuous tenses
Once however, does not really get used as a preposition because another preposition is often inserted to be more specific. For example, the prepositions are in bold:
- Once upon a time
- Once in my life
- Once on the planet earth
So to conclude, there are similarities between once and since, both describe time but once is finite and since is continuous.
I always thought of after as a straight forward preposition but looking at the dictionary definition it is indeed quite versatile and can be used as an adverb as well. "After I made the cake I felt ill". In this context it would mean "immediately after".