How long is the exam? there are creches that take children for a couple of hours while their mothers do the shopping on saturdays (I know that some nurseries in the area become creches during the weekend) Would that help?
Talk to your institution as soon as you can. Arrangements will be in place to help people who are unable to take the exam at the normal time. Jewish students can't take exams on Saturdays, for example, and there are usually special arrangements made for them to take the exam before the beginning of the Sabbath. However, there are usually complicated procedures in place to keep the people who've taken the exam early 'in isolation' so that they can't pass on information about the questions to the rest of the class. This might not be appropriate in your case, since clearly you have to look after your child(ren). If the exam has just been changed, they haven't given you long to arrange alternative childcare. Sorry, I don't know how old your child or children are, but I do know of people who have done exams with tiny babies actually in the room with them (they put them in a separate room from everyone else, obviously) with breastfeeding breaks! If your children are older, do you know any fellow-students taking different modules that you could pay to look after them for a couple of hours? But definitely ring the exams office of your institution in the first instance. I think that the onus should be on them to sort this out for you. Good luck.