no need to register as limited co or limited liability partnership, just as a partnership here.
Each partner also registers for self assessment.
Broadly speaking, the partnership return says that the partnership earned say £30k - remember the partnership expenses to include (toys, paints etc ) used in child-minding.
The partnership return, from memory, asks for the split of the profit for each partner, say 50:50.
Each partner then states their self employed income as £15k, and fills in the rest of the form.
You will need to keep reciepts for all expenses and all paperwork for 7 years.
Tax and class 4 NI is paid in 2 lump sums, at end January and end July.
Class 2 NI is paid weekly or monthly.
In my experience, if you put a third of your income into a savings account, this should cover tax and NI bills when due.