Yes there are warnings with Julia 1.3, but things should still work. Your error likely comes from the server not responding fast enough, so that eglot thinks it’s dead.
There’s an entry about this in the FAQ of eglot-jl. I would recommend cding to the project that you want to work with, and running the following command in a terminal:
julia --project=path/to/eglot-jl/ path/to/eglot-jl/eglot-jl.jl
It will probably take a (very) long time, but eventually you should get the message:
[ Info: Received new data from Julia Symbol Server.
Unfortunately, we don’t have (yet) any user-friendly way of measuring the progress of this indexation process. A patch has been incorporated in LanguageServer to log progress information to the terminal, but I don’t think it has been released in a stable version yet…
[EDIT] not so user-friendly, but you can look for files appearing in ~/.julia/packages/SymbolServer/vTGsB/store/ to get a sense of how fast the indexation process goes.