No it is not up to you or your nanny with regard to the tax side of things. If you pay above the NI LEL (currently £107 per week) then you are legally obliged to operate PAYE. See Employer's Responsibility to Operate PAYE - HMRC.gov.uk
The FT reported in March that HMRC were going to crack down on nannies being paid cash-in-hand. So the days of parents not operating PAYE for a nanny are probably quite limited.
Regarding Salary...
To me, if it is for a full-time nanny, say working 50 hours a week, then I feel it's a bit low... if you want an experienced nanny - I'm in Surrey, outside of London area and work 40 hours a week for a little more than you are proposing salary wise. How many hours per week would you be needing someone to work? Keep in mind that you probably need them to arrive before you leave for work and that the get to leave after you have got back from work. If your DH leaves for work later than you do, or gets back sooner, then that may well help cut down the number of hours of childcare you need a nanny to do. I finished work at 3.30pm today as one of the children's parents is able to take over at that time.
If you are happy to have someone new to the job role, then you may well be able to recruit someone.
You can look at other job ads in your area to get a feel for local salaries, not just nanny jobs but also in other work sectors, including nursery work. Sites like www.greatcare.co.uk have listings of Nursery jobs, sites like www.nannyjob.co.uk has nanny job listings. It can be tricky to work out what salary is on offer in some of the job ads, and some don't even give a salary but with luck you will get some idea.
I would suggest that you calculate what you feel you can afford to pay, then offer a bit below that... so you have room to increase salary over time, such as following probation, following end of year 1. Keep in mind that there can be various costs involved beyond just the salary - such as activites, travel and food whilst on duty.
MrAnchovy's PAYE calculator will help you get a feel for tax/ni and employers NI is due on a Gross salary, plus do a Net to Gross calculation if needed.
Your 40,000 assuming a 810L taxcode, is 29,732 takehome.
A gross salary of 20,000 would cost the employer 21,728 once adding on Employers National Insurance.