Most payroll providers will provide a contract... some will provide a template, others will create a bespoke contract to your requirements. This will come at a cost, either as an optional addition to the payroll service, or be incorporated in the annual fee.
Some companies will refund you for months not used subject to a minimum period. NannyPaye for example would refund 6 months if you were not going to be using a nanny again... or you could keep that 6 month period as a credit on your account to use in the future. In theory I think you could carry over the credit for several years - so if you only had a nanny each summer (say using July, Aug, Sept in terms of payroll) then a 12 month subscription could last you 4 years. You would need to check this with payroll companies that offer refunds/credit for unused months.
All payroll companies will initially want you to pay for a years worth of payroll admin. Some companies will charge you (or reduce your subscription by a month) each time you make a change - such as a change in hours, change in nanny, change in anything really.
One payroll provider handily provides a list of what they consider are chargeable changes:
A Change means any temporary or permanent change to:
an employee’s pay,
tax code (apart from the change at beginning of tax year),
name,
address
change of paydate
pay rises,
pay cuts,
changes in hours including overtime,
unpaid leave,
bonuses,
changes of name and/or address,
the starting or stopping of childcare vouchers or variations to the notified amounts to be paid by vouchers
termination of an employee’s employment
engagement of a replacement employee.
When looking around at payroll companies, do watch out for any charges for Changes. There can be many changes in the usual running of a payroll. NannyTax and NannyPaye are the largest payroll providers and they both do fully inclusive, not charging for every little change. Smaller providers can often have a lower headline subscription figure but read the fine print to check what they do and don't charge for in terms of making changes.
Pensions for small employers starts soon and for some has already started. If this if the first time you are to be an employer, you won't need to provide a pension scheme until November 2017, so I would not worry about that yet.
With regards to if income tax, national insurance and employers national insurance would be payable, it depends on what your nanny is paid and their personal financial situation. If they are able to just work Summer Term, are they a student at the moment? They may or may not have had some other work, so that can make a difference to tax calculations. The payroll service will sort all that out for you. It can take several weeks to initially set you up as an employer with HMRC, so do get in touch with a payroll provider before you need to start paying your nanny. The more details about your nanny you have the better, though obviously you may not know lots of detail until such time as you have offered them the job.
Hope that helps.