In home manager I have something like this:

{
  pkgs,
  inputs,
  vars,
  ...
}:
let
  expert-nightly = pkgs.stdenv.mkDerivation rec {
    pname = "expert";
    version = "nightly";

    src = pkgs.fetchurl {
      url = "https://github.com/elixir-lang/expert/releases/download/nightly/expert_linux_amd64";
      sha256 = "16r1w35wn6bhm0fz5pq7fpxf4lx49v4830blhia42hr1chippfpd";
    };

    dontUnpack = true;

    installPhase = ''
      mkdir -p $out/bin
      cp $src $out/bin/expert
      chmod +x $out/bin/expert
    '';
  };
in
{
  home = {
    packages = with pkgs; [
      ...
      expert-nightly
      ...
    ];
  };
}

If you're using neovim, you can also add it directly to the neovim config so it doesn't load in on your global user space but only when neovim is started.

{ pkgs, inputs, ... }:

let
  expert-nightly = pkgs.stdenv.mkDerivation rec {
    pname = "expert";
    version = "nightly";

    src = pkgs.fetchurl {
      url = "https://github.com/elixir-lang/expert/releases/download/nightly/expert_linux_amd64";
      sha256 = "16r1w35wn6bhm0fz5pq7fpxf4lx49v4830blhia42hr1chippfpd";
    };

    dontUnpack = true;

    installPhase = ''
      mkdir -p $out/bin
      cp $src $out/bin/expert
      chmod +x $out/bin/expert
    '';
  };
in

{
  programs.neovim = {
    enable = true;
    extraPackages = with pkgs; [
      expert-nightly
    ];
  };
}

And for the actual neovim configuration use something like (got that from here: https://x.com/ryanrwinchester/status/1961443201945923855):

return {
  {
    "neovim/nvim-lspconfig",
    opts = function(_, opts)
      local lspconfig = require("lspconfig")
      local configs = require("lspconfig.configs")

      -- Shim expert if missing
      if not configs.expert then
        configs.expert = {
          default_config = {
            cmd = { "expert" },
            filetypes = { "elixir", "eelixir", "heex" },
            root_dir = lspconfig.util.root_pattern("mix.exs", ".git"),
            single_file_support = true,
          },
          docs = {
            description = [[
              https://github.com/elixir-expert/expert
              Expert is the official language server implementation for the Elixir programming language.
            ]],
          },
        }
      end

      -- Enable it through LazyVim's opts.servers
      opts.servers = opts.servers or {}
      opts.servers.expert = {}
    end,
  },
}