You can install Signal without gargle play, but for some reason it’s not easy to find from their index page.
Incidentally, I came here looking to see if anyone else was having the same problem updating Signal as me right now, with the update notification doing nothing, and I might have to download this directly myself, but it seems to be not the only non-f-droid app that’s not updating properly at the moment.
Also, I had the same problem as you just described back when I installed whatsapp before giving up on it. With this pattern, I wonder if anyone can work out what’s going wrong here…