In these contexts, to get on [well] means to be successful, content, whereas not getting on means failing (or otherwise being unhappy with one's circumstances).1
1: Let me know how you get on
...is perfectly natural English phrasing (that's a link to hundreds of published instances in Google Books). But note that #1 above ask to be kept informed of the addressee's future state (regarding some anticipated future situation). But the question format...
2: How are you getting on?
...asks how things are for you currently.
We don't use "explicit future" syntax in contexts like this, so...
3: Let me know how you will get on
...is (idiomatically, if not syntactically) invalid. The only contexts where the first 6 words of #3 above are those asking about what method the addressee will use to accomplish something, such as...
3a: Let me know how you will get to the airport
3b: Let me know how you will inform the next of kin
...where in both cases it's possible the addressee can answer immediately (because he already knows how he intends to get there, or to tell them). But that question format can also be used if both parties are aware that the addressee doesn't yet know (or hasn't decided) how the planned future action will be accomplished.
1 Note the special case where the addressee (you) in example #2 above is plural (speaker is addressing two or more people collectively). In that situation, How are you getting on? is ambiguous. It might be asking whether the addressees collectively are "succeeding" in some current task, but it might be asking whether they have good personal relations with each other (are they getting on/along well together?).