Setting all the emotions & touchy family dymamics aside aside for a sec, isn't the OP legally entitled to 1/3 of his fathers estate? His father died without a will. Intestacy laws therefore apply. Under those laws, the surviving spouse gets 2/3 of the deceased estate, with the remaining 1/3 going to the children. What the mother decides to give or not give her son, or what the son feels he may or may not be entitled to should be irrelevant. The law sets out in plain black and white who gets what.
The only exception to this is if a bank account was held by the parents jointly, in both their names. If that was the case, the bank would transfer over the funds to the other account holder on the production of death certificate. No other parties have a claim on it. But if it was in the fathers name only, they should not have transfered anything to anyone unless they saw a will or a Grant of Probate or Letters of Administration giving that person legal entitlement to the funds.
OP, if I were you, I'd approach the Probate office and see what they can tell you about his estate & were letters of administration ever taken out, by who and when. Then pop into the bank and find out if you can what the status of the account was, joint or in your dads name only. Or get a solicitor to write to them to find out. You may have had a legal claim to funds that they had in their care. They can't just brush you off without potentially getting themselves into trouble.
If it turns out the account was a joint one, let it go. The monies in it are your mams now. She is and was legally entitled to all of it. She has zero legal obligation to tell you or anyone else anything about it, or give you a single penny of it. If she has gifted you money since then, she is under no obligation to tell you its source. It would be nice if she did to put your mind at rest, but she has no legal obligation to do so. That may tick you off, but its just how things are, let it go.
If the account was in your dads name only, then you have to decide what you want to do, let it go, or go after your mother for what you are legally entitled to, and risk harming your relationship with her forever. With all due respect to your mother, her opinion that she didn't "believe" in wills would not be the most educated or enlightened of viewpoints imo. Perhaps she is just not aware of what happens when someone dies without a will. Perhaps she just presumed that everything automatically went to her when her husband died, when in fact it should not have. It's up to you to decide how best to to approach her with that fact, if at all.