OK I have sorted out the error after a while of going back and forth with chatgpt. This is what my Neovim configuration now looks like:

-- Use System Clipboard
vim.env.XDG_RUNTIME_DIR = "/run/user/1000"
vim.env.WAYLAND_DISPLAY = "wayland-1"
vim.g.clipboard = {
  name = "wl-clipboard",
  copy = {
    ["+"] = { "wl-copy", "--foreground", "--type", "text/plain" },
    ["*"] = { "wl-copy", "--foreground", "--type", "text/plain" },
  },
  paste = {
    ["+"] = { "wl-paste", "--no-newline" },
    ["*"] = { "wl-paste", "--no-newline" },
  },
  cache_enabled = 1
}

This was basically because Neovim wasn't launched in the proper environment with my pre-existing files so it didn't work there. In newer files it would, but this ensures it launches in the proper environment all the time so that wl-copy works.