Sharing of resources sounds quite a nice idea to me. Your friend is happy with it so give it a go and see how it pans out. If you find that it isn't a fair exchange, then you could always find something else to add - for example you buying a weeks food once a month.
How old are the children?
While you don't use a car when taking out your DD, what about if you took your friends two children as well - is taking 3 children on public transport going to work? I find taking two children on public transport works, but once you add a third it can be more challenging - it depends on their ages... the younger they are, the harder it will be. Given you say you live near everything... I guess you live in London - as near is a relative term... things near me tend to be 7-15 miles!
As a nanny I've never had a mobile phone as part of my job. I don't think it would be usual - though may be for a live-in nanny who is coming to the country from abroad.
If you are wanting your nanny to frequently keep in touch with you via mobile phone, then paying towards the cost of a contract phone may be viable. You may want to look at tax implications - may be easier to get them a phone, so you get the itemised bill.
As a nanny I don't do much cleaning... tend to wipe surfaces in the kitchen, load/unload dishwasher, try to keep the babies highchair in a reasonable state, run the vacuum around on occasion. Tidy the children's bookcase - especially when littleone pulls most of the books off. I do some of the children's washing, which is an ever ongoing task - my boss is nice as she will try to keep the children's washing up-to-date as well, so all in all we usually have sufficient clothes for the children at all times (though the sock monster does seem to be feasting on socks at the moment).
In a share doing the children's washing when all care is based at one home will mean the washing for that family (your friend), not the share family (you). However, your DD will need a supply of clothes at your friends home... so those clothes could go in the wash there, as and when required. Your DD having spare clothes at your friends home I feel is vital, even if you do live close. It is a lot easier if there are spare items that can be used when the children get mud/paint/food/snot/puke etc over their clothes.
Sorry for telling you to search out message threads earlier... I was on a mobile device so could not post links to useful things. Also much easier to answer specific questions rather than one that is more general.
As NannyL says, your starting point is to plan everything out. You need to identify your needs, thus creating a job description, which will then lead to creating job advert(s).
Recruiting yourself is probably the best thing to do, if you are not in a hurry. www.nannyjob.co.uk is the most popular job site for recruiting a nanny in the UK, in my view.
If you are not London based, then also putting an advert in a local section of www.gumtree.com can also be useful, to help get more applicants. Also consider local advertising - newsagents window for example, school newsletter.
You need to agree everything with your friend, have the same views on things such as children's behaviour, what children eat, where nanny can take children. You need a contract between yourselves, which sets out the key things about the share... you may then also want to add some kind of Handbook to that contract - which you can keep updating as and when you find new things you need to discuss and agree. When problems do arise you need to sort it quickly - have a weekly meeting where you and your friend chat about how things have gone that week, any issues, and agree on how to sort it out... all over a glass of wine, coffee, or a meal. Consult with your DH's, keep them involved to however much extent they wish to be involved (some will want more involvement than others).