As I wrote on your other thread, there is no legal requirement for it to be signed.
I shall expand on that for you:
Under The Employment Rights Act 1996 employers have to provide to their employees, within 2 months of their start of employment, a written statement summarising the main particulars of employment.
The written statement is evidence of the contract of employment - a legally binding agreement between the employer and employee which is formed when the employee agrees to work for the employer in return for payment.
So if you have accepted payment which I shall assume you have done, you have a contract between yourself and your employer. The written statement sets out that agreement, though in reality you may not always work exactly to what that statement says - you employer may let you leave early, your employer may let you take more holiday that it states, the working hours may change a bit though the number of hours per week may stay the same. Ideally these changes should be documented though often when working as a nanny they are not, instead it is talked about and sometimes put on an email.
What is your actual issue? You are wanting to leave the job, so I presume that the problem is to do with notice period that you need to give your employer. How long is the notice period and do you view it as being unreasonable?
Once you have given notice your employer may find a replacement quicker than the period stated, so you might agree at that stage to part company as it is mutually beneficial to both of you (presuming you are wanting to leave earlier than the notice period in your contract). However it may take your employer quite a while to find a replacement.
You will want a reference from this employer I would guess, so leaving on as amicable terms as possible would be good. If your new employer really wants you, then would they not be prepared to wait until you can start working for them?