I use a bit of duct tape on this kinda injury for day-to-day protection – really just so it doesn’t catch on something and pull it the wrong way (which it will).
😯
This description might not work in narrative 🙄 , but … cut the tape into an H shape, fold the inside bits of the H in on themselves, so the stickiness is covered by tape, and put that bit over the injury – use the unfolded ‘arms’ of the H shape to stick to the rest of your finger – this’ll save the tape ripping the damaged bit nail when you change the tape. Other than that it’ll sort itself out (eventually).
Have a look online for ‘field-expedient duct tape dressing’ or some kinda variation on that. You can reliably hold all sorts of injuries together like this.