Both sentences place the collection of the money in the past:
... who collected the money.
The past tense "I was the one" refers to the situation in the past. For example:
Cousin Joe and I were drawing lots to see who should collect the money. I drew the shorter straw, so I was the one who collected the money.
The present tense "I am the one" refers to the current state of affairs. You are the person responsible for carrying out that action, and your responsibility extends into the present.
I am the one who collected the money. If there are to be any consequences for that action, I am the one who must suffer them.
P.S. But I do not really understand what "to collect mummy's money" means. Does it mean "to remove the money surreptitiously from her purse"? Or does it mean something like "Went to a person for whom your mother did some work, and asked for payment"? Or "to gather up some loose coins or hers, from her desk, for example, or perhaps some paper bills from the pocket of a blouse in the laundry basket"?