I would start by getting an attorney involved asap. Usually a letter will from an attorney will help remedy the situation. If not, unfortunately the only option you have is to take him to court.
The unsigned will has no force and effect, but you, as a child, have the standing and priority to go into court and address this if needed.
The will is ineffective so all property passes through laws of intestacy. As your mother's boyfriend refuses to cooperate, your first step would be to contact a probate attorney for your options. Once the boyfriend knows an attorney is on board, the situation may improve quickly.
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You should contact an experienced trusts and estates attorney near you, and soon. The unsigned will has no effect. Your mother's property will pass to her heirs at law, her children. Whether or not the boyfriend is entitled to the accounts will depend on whether they were joint accounts held in both of their names, or if whether the accounts had "pay on death" instructions and he was named as the payee.
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See a Probate ATtorney immediately and get a Petition to probate the estate filed immediately. The Probate attorney will know what to do regarding the boyfriend. No one can do anything without court authority.