The workaround is to use the following script command on the REPL shell mode:
shell> cmd /C c:\\cygwin64\\bin\\bash -i -l
oheil@PC ~
$
I can only speculate: Julia is spawning commands directly using exec(), without any underlying terminal, not like e.g. running a remote command using ssh on linux/unix where the command is run with a shell (/bin/sh) as the execution part. This difference (missing terminal) is perhaps the cause for this weird behaviour above.
The workaround uses windows cmd to run the cygwin bash. Options -i -l are there to have have bash as a interactive login shell (e.g. to have typical commands like ls in your path).