Do I need to add tax on to the basic hourly rate?
A Gross rate is BEFORE employee tax and NI deductions. You would need to add Employers NI.
A Net rate is AFTER all deductions. This is not recommended, especially for part-time nannies, as their personal tax allowance could all be used in another job.
calculator.kistax.com will give you an example of the figures. A payroll company - there are many, www.payefornannies.co.uk for example will do a lot of the paperwork for you, for a fixed annual fee.
What is the average rate for West London nanny?
London rates can be fairly high, I'd say £9-12 gross per hour would be typical for a qualified nanny... with higher end of range being for very experienced nannies. A newly qualified straight out of college may accept a lower wage, best to work out what you can afford to pay... then offer a bit less, so there is room to give a payrise later on.
Where can I advertise/find a good nanny?
www.nannyjob.co.uk
Childcare.co.uk
Advertise on both sites... putting as much information as you can in your advert, especially covering things like working hours, working days, a salary range, expected duties - some nannies will expect to do some child related housework, others won't. So be clear as to what you would expect. I don't Iron (as I'm useless) but I do load/unload the washing machine and dishwasher, push the vac around on occasion. Make sure you specify if you want a Driver.
As a Dr you probably won't get childcare vouchers (unless you are an employed Dr, then you might) but do check with your partner if they can get them, as Childcare Vouchers can be used with a Registered Nanny. If you will be using vouchers to part-pay the nannies salary, specify in the advert that you want someone already registered.
Consider how far someone is travelling to the job. Reliability is very important, so you don't want someone who will be travelling quite a long way in London traffic.
Search this section of mumsnet for lots of useful info about things like Contracts, expected duties, potential issues/problems. You can use Advanced Search to search using a keyword and limit it to just this topic.