The algorithm used is called CHIRP by Dr Bouman from CSAIL. I guess I oped too much that it might be written in Julia! I guess not…
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Details of the data processing are in one of the 8 EHT papers just released (https://iopscience.iop.org/article/10.3847/2041-8213/ab0c57). Looks like a lot of what they use is Python based with numpy and scipy extensions plus some heavy duty crunching packages which might be written in anything.
Half a ton of hard drives
In the paper there is a link to a GitHub repository, where is a tool used for the analysis: GitHub - sao-eht/eat: EHT Analysis Toolkit However there doesn’t seem to be much care about the code, it doesn’t even have a licence.
Not yet. 
Seriously, the chain from telescopes to image is very complex and for sure I know that FORTRAN, C, C++, and Python are used, maybe more. As you can learn from the papers, multiple algorithms (written in different languages) were used and compared to produce the final image. Julia may play a role in some stages of the chain in the near future.
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How to Understand the Image of a Black Hole
This is better maintained: https://github.com/achael/eht-imaging