Yes, her eldest child would be next of kin and entitled to the ashes, assuming there was no will.
The starting point for a court in this dispute is to consider whether ashes are "property". Following the case of Williams v Williams , there is no property in a corpse.
The law states that if there was a valid will, the executor is entitled to the ashes and to decide what to do with them (cremation or to scatter, etc). The primary duty is for the executor, in other words.
If there is no valid will then the rules of intestacy will apply and in that situation the highest-ranking next of kin would be entitled to the ashes, which would be the deceased's Husband in this situation if she was married, or her children who are next, then it would be siblings (if there are no children).
This is confirmed in Section 22 of the Non-Contentious Probate Rules 1987. It depends therefore if there was a valid will - if there was, the executor will have the right to the ashes and the executor could be someone other than the surviving spouse - it depends who is nominated in the will. If there is no will (or no valid will), the next of kin will be entitled to them.
You may need a law firm to help with this if the partner refuses to change their stance.
Here is a list for you :