"But if a parent dies intestate, wouldn’t the remaining parent become a trustee of the children’s money that’s left by the deceased parent?"
If they were still married then the surviving spouse would indeed become the trustee (the term used is 'administrator').
However, they are no longer married and ex-spouses don't count.
The order of who can apply to be administrator when there is no will is
1 Spouse or civil partner
2 Children over 18
3 Parents
4 Siblings
5 Half-siblings
6 Grandparents
7 Aunts/Uncles
8 Half aunts/uncles
so, since he is divorced (I assume?) then he has no spouse, his children are under 18 so they can't do it either.
Next come his parents, if either of them are alive then they get first refusal for administering his estate. After that it would be any of his brothers or sisters.
So, if your DP were to pass away without making a will while his children are under the age of 18 then it would be his parents that would be looking after the money from his estate.
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"So realistically, anything left to the children without a clear will stating she is not to use it, can be accessed by her."
No, this is totally wrong. An ex-spouse will not get their hands on anything.