Alacritty doesn’t do well with modifier keys and a lot of keyboard shortcuts don’t work in neovim by default. I recommend using foot instead.
All tmux and fish keyboard shortcuts have worked without any alacrity configuration (that I can recall).
My tmux config adds shortcuts to behave like a conventional tabbed window without any issues e.g. ctrl-n (new tab), ctrl-w (close tab), ctrl-page-up/down to navigate tabs.