English questions. Please help explain the meaning of “nearly” and “barely” in the following sentences.?

"Nearly" means "almost did"... "barely" means "almost did NOT".
The sentences mean the opposite of each other, partly because of the "nearly/barely" adverb, and because you have chosen opposite-meaning main verbs: "remember" means the opposite of "forget".
Therefore:
The first sentence means she did NOT bring the wallet.
The second sentence means she DID bring the wallet.

The grammar in the second sentence is incorrect. "She" is third-person singular, so the verb should be "forgets" - not "forget". Furthermore, "forgets" is correct only if this is a current, ongoing situation. If, as I suspect, this is a PAST situation, it should be "forgot".

Finally, "barely forgets" or "barely forgot" is neither natural, nor is it a phrase that would be commonly used. "Nearly remembered" is natural, but would not ordinarily be used. Both mean that she forgot - which is what we would say, without an adverb. "She forgot to bring the wallet."

The following would be more natural and usual, and they mean the same thing:
"She barely remembered..." means she ALMOST did NOT remember - but she did; therefore she did bring the wallet. For example, "She was doing five things at once, because she was short of time. She was so busy, she barely remembered that she was supposed to bring his wallet. But, at the last minute, it popped into her head. She DID bring the wallet."

"She nearly forgot..." means she ALMOST forgot - but she did not; therefore she did bring the wallet. For example, "She was so busy, she nearly forgot that she was supposed to bring his wallet. But, at the last minute, it popped into her head. She DID bring the wallet."

Neither of those would be used. In both cases she forgot, barely and nearly don't really mean anything.
You could use both of these expressions the other way. "She barely remembered" or "nearly forgot" would indicate that she remembered at the last minute.

she nearly missed the bus.(but did not)
she barely makes enough to meet ends.(just about)

She doesn’t have a good memory