3-Day per week nannies certainly exist. I've done a nanny job that was 3 days per week in the past... did nursery temping on other days.
AskNannyNick has some info for parents thinking of employing a nanny, such as paying a nanny (PAYE), childcare vouchers, recruitment, holidays etc.
You will also find a lot of information on this section of Mumsnet (advance search can be used to limit search of Mumsnet to just this section).
The cost of three under 5's places at nursery or at a childminders could well be more than the cost of a nanny. Also getting those places may be tricky, as a CM is often limited to 3 under 5 places. However one big advantage of a nanny over other forms of childcare when you have 3+ children, is that the nanny comes to your home... you don't need to get 3 children up, dressed, fed first thing in the morning. There are negatives of course, such as you will be an employer and need to operate PAYE.
Where do you look?
www.nannyjob.co.uk is the most popular website for nannies looking for jobs in my view. An advert there costs around £20 for a month.
Local advert in a newsagents window can work, as can word-of-mouth... ask around at toddler groups for example.
how much does it cost?
Salary can vary, £7-£12 gross per hour, dependent on your location. To that you add Employers NICs (see details on PAYE).
There are then other additional costs such as a weekly petty-cash kitty for nanny to use to entertain your children, the cost of food (nannies get food whilst on duty), heating/lighting as your home will be used during the day. If nanny uses their own car, then mileage payments are 40p per mile for first 10,000 miles per year.
any other practicalities to consider?
How will nanny transport 3 children, particularly 3 under 5's who all need car seats. Fitting 3 car seats in a small car can be fun. Travelling with 3 children on public transport can be a lot of fun!
Buggies - I'd suggest making the older two children walk. Far easier having a single buggy rather than a double.
Sickness - children can be sick, a nursery or CM won't care for sick children... but a nanny does. Depending on the seriousness of the illness, you would want to care for your sick child yourself. Nannies themselves get sick - so be prepared for using some of your holiday entitlement from work to cover nannies sick days (many nannies seem to have very few days off sick, so don't panic about this).
are there registered bodies, checks etc?
In England, Nannies can be registered by Ofsted (Ofsted registration means that you can use Childcare Vouchers and means that the nanny meets a minimum standard). Guide to Ofsted Childcare Register (Nannies)
Nannies should have an Enhanced Criminal Record check - this is done as part of the Ofsted registration. Nannies not Ofsted registered can get this check done via a Nanny Agency. In 2010 there is to be a new system called Independent Safeguarding Authority who will be doing continuously updated CRB checks... all childcare workers will have to be on the ISA register, though the date when it becomes compulsory may be many years away (suspect it will remain voluntary for nannies for a long time).