There will be no change to the rights and status of EU nationals living in the UK, nor UK nationals living in the EU, while the UK remains in the EU.
Gibraltar is a British Overseas Territory. There’s no formal British consular representation in Gibraltar and the local authorities deal with all requests for consular assistance. See Consular assistance
You may experience delays when entering and exiting Gibraltar by road or on foot. See Road travel.
Terrorist attacks in Gibraltar can’t be ruled out. See Terrorism
Most visits to Gibraltar are trouble-free. Violence and street crime are rare. See Crime
If you need to contact the emergency services call 199 (police) or 190 (ambulance and fire).
Take out comprehensive travel and medical insurance before you travel.