Some night nannies will be booked up early especially good ones in london. Where I am there is not a huge call for night nannies so I tend to get one longish job a year then lots of ad-hoc nights where parents just want a nights sleep or couple nights sleep with some routine advice they can follow.
If you use night nannies the agency you will pay an agency fee then the nanny £8-£9 an hr.
Privately you would be looking at paying £10-£15 an hr depending on experience and qualifications.
The experience you want from a night nanny depends on whether you literally just want the sleep so nanny comes in and basically takes over for the night - therefore you could use someone who has lots baby experience looking for first night nanny job and maybe pay around £8 an hr for this.
or if you want someone who can advise and implement routines not just overnight but give you routine advice for the daytime then you want an experienced night nanny and would be looking to pay £12-£15 an hr.
You want someone you warm to immediatly you meet them so you feel relaxed enough to actually sleep when they are there.
Make sure you get references 2-3 recent ones would be recommended. crb checked, personally would want non-smoker around a newborn baby.
Are you going to breastfeed if so what are you going to do at night, express? nanny bring baby into you? Personally I suggest the mum does the 10pm (ish) feed as a breast feed then expresses for the 2am (ish) feed and then at 6-7am nanny takes baby into mum for feed.
Nanny will take care of all babies needs changing, feeding, winding etc. If using bottles I will wash them out and pop in steriliser if enough bottle to put it on and obv if I have time as baby comes first. Some night nannies will do some baby ironing when they arrive til baby needs dream feed I personally dont as I hate ironing with a passion but will fold clothes, get out of washer if needed.
Most night nannies work 9pm-7am and will do 1-7 nights a week. I tend to only do 2-3 nights a week split over mon-fri - always doing a fri then parents tend to like sat/sun to themselves as a family.
Hope this helps.
I would suggest www.netmums.co.uk for finding a private night nanny. Could try www.gumtree.com but be prepared to filter out all the unsuitables (rubbish!!)