The OP's original thread from June is here:
https://www.mumsnet.com/talk/legal_matters/5349355-grandfathers-will-testify
"The trustees granted me inheritance money from my father."
In the original thread you said that your uncle was the executor of your grandfather's estate. His estate needed to be distributed according to the grandfather's will which left some money to your now deceased father.
I am confused as to who these "trustees" are?
"Do I need to apply for the letters of administration now?"
Did your father leave a will or did he die without making a will?
If he left a will then whoever is named as executor in the will can apply for probate. If he didn't leave a will then it will be his nearest living relatives who can apply for letters of administration. Assuming that he wasn't married when he died then that would be his children (ie you).
Has nobody applied for probate or letters of administration for your father's estate yet?
If not, then that is the first thing you should do.
"This has caused a reconsideration meeting going ahead."
Again, I'm confused by this. The executors of your grandfather's will have no choice in the matter, they have to leave whatever it says in the will to your father's estate. They cannot hand it on to other random people.
It is then up to the executor of your father's estate, or the person who applied for letters of administration to administer his estate.
"However a third party who I know who this is has appealed the decision."
Again, this is all very confusing - are you in England & Wales or a different country?
It sounds as though your father was living with somebody at the time of his death.
Depending on their circumstances and any evidence that she has, she may be able to bring a court claim against your father's estate. She would have to show that your father was financially supporting her at the time of his death.
She has six months from the date that you get probate for your father's estate in which to bring a court claim.