Depends what the arrangement is. If you are having the nanny exclusively for say, three days per week and they are having her for the other two then you will each have a separate contract with her and they should make their own arrangements in relation to payroll, insurance, etc.
If she is looking after both children at one or other house, even if it is a short-term arrangement, it would be wise to put in place a contract that deals with this, setting outwhat proportion of her salary each of you is liable to pay and confirming that you will each deal with the tax on the relevant proportion of her salary. You don't want to be left out of pocket because your sharers don't reimburse you for salary which you have paid and you want to ensure that all tax is properly accounted for because if it's not and you're the only registered employer, you may find that HMRC tries to come after you for the tax.
Depending on the duration of the arrangement, you also need to establish how you deal with holidays (i.e. does the nanny have complete discretion about when she takes them, or do the two sets of parents each have a right to ask her to take a certain number of days). I'm assuming that as she's primarily your nanny, you'll retain the ultimate discretion to fire her if she does anything bad, but again, the contract should deal with this.
As regards insurance, if you are both employers, you'll both need employer's liability insurance. You should also check that your home insurance covers any injury to the other child whilst he/she is in your home (generally, it does, up to certain limits).
It all sounds rather formal for a temporary arrangement, I know, but it's probably better to set out everything at the outset rather than keep the arrangements loose and have a dispute later. It also lets the nanny know where she stands.