We haven't measured the actual time. I would say the time learning Elixir has not been a hurdle or anyway limiting factor to the projects we are working on.
We basically switched as a company from writing servers in C#/nodejs/python to just Elixir. Did this gradually during end of last year and this spring.
Though I must mention that all of our developers were already familiar with multiple languages. And since we have multiple projects going on we can transfer knowledge easily. So YMMV.
I love the Elixir syntax but worry it'd be like Scala: a better way for people who already know Java, rather than a way to use BEAM / OTP etc for people used to modern languages.
There are a handful of rules to remember, if that, and it's reasonably straight-forward. module:func becomes :module.func, the erlang func probably wants char lists rather than strings, atoms are lower-case and should be changed to :atom, vars are upper case and should probably be lowered.
Learning the Elixir language is easy, thinking functionally less so (for me at least).
Agree with rossj, there're some rules to remember and then it's not that hard.
Learning the procedural aspects of Erlang or Elixir is very easy. In that sense I agree with the previous responses. Erlang gets a lot of bad rap for the syntax. But, it is really no more difficult that any other language to learn. The syntax can easily be learned in a day. Elixir's syntax can be learned even more quickly because it looks a lot like Ruby or Python. Thus, you can certainly start write scripts in Erlang or Elixir relatively quickly. My guess is that the average, experienced developer could start writing basic scripts in an hour or two. However, those scripts would be entirely procedural in nature, unless they are already very familiar with functional programming and pattern matching.
The first hurdle in Erlang/Elixir is the functional aspects of the language and understanding the power of pattern matching. Both of these require rethinking how you build a program. But, both of these are also incredibly, incredibly worthwhile and powerful. Elixir in particular is a great medium for better understanding these concepts because it is a very approachable language. Getting over this hurdle did not happen over night for me. I rarely used recursion in Ruby or Javascript (there is no tail call optimization). On the other hand, in functional languages recursive functions are the norm. I learned to really love them with Erlang/Elixir and now it frustrates me when I can't use them elsewhere because it has become much easier for me to reason about recursive functions than loops. Overall, functional programming and pattern matching are amazing things to learn but really grokking them won't happen over night.
The next hurdle with Erlang/Elixir is understanding OTP. This is THE incredibly powerful aspect of the Erlang VM. However, it is also not an easy subject matter to approach unless you already have a strong background in distributed systems. I did not. Hence, it took me a while to compose programs that actually used the Erlang VM for what it was built for, OTP. Thus, this portion of the Erlang/Elixir learning curve felt long and slow. I was not used to spawning new processes on a whim and passing messages back and forth. Initially I would send messages only one way without realizing that the appropriate pattern is to view it in a standard server-client relationship where there should always be a request and a response. Eventually I started viewing all the little applications running inside my larger program as self-contained capsules/servers that communicated among each other. However, it took me weeks of really intense study to get comfortable with. But, once you get comfortable with OTP you have a lot of built-in power (people rarely mention Mnesia, DTS, ETS gen_server but they are awesome).
Overall, my experience with Erlang/Elixir has shown me one big thing: pick the best tool for the job and there are a lot of jobs that you don't need OTP for. When you need to build Whatsapp, you should pick Erlang/Elixir. When you want to build a simple web app that only has a couple of end points and much of the dynamism is handled on the front-end, use Go, Node, or my personal favorite Clojure. Erlang/Elixir shine in the context for which they were built. But, they are not built for everything. And, if you only want the procedural aspects of the languages, there is no reason to choose them over Go, Node, Clojure, or Scala...all of which have solid approaches to concurrency.