You'd still use the word 'kecemasan' for these examples:
The hospital will cater only for emergencies. - Hospital itu hanya memenuhi hal-hal kecemasan.
Hospital Emergency Department - Jabatan Kecemasan Hospital
Emergency Room - Bilik/Wad kecemasan
Fun fact: In Indonesia, kecemasan means "uneasiness/worry". For the word emergency they use the word "darurat". In Malay however, "darurat" has a specific meaning which is "a state of emergency".