Not sure how it works in Nursing but I'm a HCP and we have an ongoing staff situation (as does 99% of the NHS)
If a member of staff leaves , they give their notice ( 4 weeks for clinic staff, 12 weeks for manager)
The funding doesn't come back to our dept, it goes in The Pot.
To fill a vacancy takes months. Post advert. Give a closing date. Send out interviews (and the applicant might not want to tell their employer that they're looking so might need to take leave- so notice needed for that).
Interviews. If they turn up 
Job offer
The applicant then gives their notice (a month)
So it can take 3 months+ (as I tell my patients)
"Oh is no-one going in for this job" they ask.
Loads but the system doesn't work like that.
IIRC , if staff leave the agency to go into the NHS, the NHS have to pay (a month salary I think) to release them.
Agency is good money (I did for 2 years) but no job security, more travel, no sick leave , no holiday, and I got the shitty clinics alot of the time.
Some wards will offer overtime to staff, but that's if they want to do another shift.
I can do overtime but as I'm P/T , i'd get time or pay at normal rate, not time-and-a-half that I would at F/T.
Also in wards, there's a legal minimum staff:patient ratio, so they will always need short notice staff.