Get it checked if the bleeding won't stop, something is stuck in it, it's a bite, it looks infected, or they're not sure their tetanus jabs are up to date.
Leave a deep splinter in and get it seen. Get help too if it looks infected (red, swollen, more painful, or leaking pus).
They need to be seen, but most people won't need an ambulance. Take them to an urgent treatment centre or A&E, or call 111 if you're not sure where to go. If you're seriously concerned at any point, call 999.