Sorry to hear of your loss. I agree largely with romulan's post, but a solicitor cannot be retained by two clients, AFAIK.
What ages are the 4 children, adults or minors?
When were they divorced, what solicitor acted for your husband, what agreements and property settlements were made and do you have copies of these? (you would have needed his divorce decree(s) to get married; they might be together).
In whose name is your current dwelling house / family home registered? Is there a mortgage in place and whose name(s) appears on the mortgage documents? If there is a mortgage, has this been discharged by a life policy with the mortgage company as beneficiary or might the unfortunate circumstances of your husband's death militate against that?
What's the story with this mediator? He is mediating between what parties, for what purpose and who appointed him? Is he a professional lawyer / mediator or a well-meaning although possibly ill-informed amateur?
While intestacy is not necessarily an adversarial situation, I feel you need to get the best family law / intestacy lawyer you can to represent your interests. I believe, as a matter of urgency, you need to be appointed the sole Legal Personal Representative of your late husband in order to guide the intestacy through probate to administration, thus ensuring at first hand that everything is done as your late husband would have wished.
BTW, just a not of caution - be careful if any detail you post might lead to you being identified.