In reply to AndyB123:
Did she make a will? The relevant phrase is actually "personal representative" rather than "next of kin". In case of family dispute then a funeral director will take instructions from anyone who is a personal representative. That is either an executor of the will if there is one, or if there is not a will, it will be a close relative excluding unmarried partners. So (if no will) her partner will not be a PR. The children will. However anyone can refuse to become an executor or PR if they wish. If everyone refuses, the local council steps in to provide a basic 'paupers' funeral as a last resort. It is obviously much better if everyone can come to an amicable agreement and a third party such as a priest may be able to help.
Did she leave any money or assets? If so, the funeral costs will come out of this pot, before legacies are paid, so there will be no financial burden to the ones organising the funeral. If not, there is a Funeral Payments Fund which may help if there is nobody who can afford to pay.