maintenance

I think both, emergency and urgency, indicate the need for immediacy. The difference is that emergency is used to refer to problematic/unwanted situations (e.g. medical emergencies), whereas urgency, though indicates the need for hurried response, does not mean a problem has occurred (e.g. urgent letter, urgent business flight).

In light of that, I guess an emergency maintenance

I would say that 'emergency' is a higher priority that 'urgent'

'The pipe is about to fail, it needs maintenance urgently'

'The pipe has failed, it needs emergency maintenance'