Neville poderia ter pego a profecia no Ministério em vez de Harry?

Provavelmente não

A profecia originalmente não mencionava o nome de Harry, ou mesmo o de Neville. Apenas mencionou algumas características dos indivíduos que poderiam garantir a queda de Voldemort, como a data de nascimento. O Guardião mudou a rotulagem da profecia depois da tentativa de Voldemort de matar Harry :

“But then . . . but then, why was it my name on the prophecy and not Neville’s?”

“The official record was relabeled after Voldemort’s attack on you as a child,” said Dumbledore. “It seemed plain to the keeper of the Hall of Prophecy that Voldemort could only have tried to kill you because he knew you to be the one to whom Sybill was referring.”

—Harry Potter and the Order of the Phoenix

Sabemos que somente os sujeitos de uma profecia podem recuperá-la:

“So he wanted me to come and get it, did he? Why?”

“Why?” Malfoy sounded incredulously delighted. “Because the only people who are permitted to retrieve a prophecy from the Department of Mysteries, Potter, are those about whom it was made, as the Dark Lord discovered when he attempted to use others to steal it for him.”

—Harry Potter and the Order of the Phoenix

Observe a linguagem aqui: são permitidos . Isso parece indicar que as proteções são encantamentos colocados pelo próprio Ministério, com base em sua compreensão de quem as profecias fazem referência. Como tal, Neville não poderia ter removido a profecia, a menos que ele fosse incluído nos encantamentos. Parece improvável que ele poderia ter sido, após a reclassificação.

Por que havia um ponto de interrogação antes do nome de Harry, provavelmente era simples prudência. Embora possa ter parecido claro para o Guardião que a profecia se referia a Harry, eles provavelmente não gostariam de fazer as coisas parecerem certas quando Harry não foi mencionado na profecia em si.

Se Lucius Malfoy estiver correto, e pelas evidências de Broderick Bode, parece que ele é,

[T]he only people who are permitted to retrieve a prophecy from the Department of Mysteries, Potter, are those about whom it was made, as the Dark Lord discovered when he attempted to use others to steal it for him.

Harry Potter and the Order of the Phoenix - p.693 - Bloomsbury - Chapter 35, Beyond the Veil

(See also Adamant's answer)

'Avery told me Bode would be able to remove it.'

'Bode could never have taken it, Master ... Bode would have known he could not ... undoubtedly, that is why he fought so hard against Malfoy's Imperius Curse ...'

Harry Potter and the Order of the Phoenix - p.516 - Bloomsbury - Chapter 26, Seen and Unforeseen

Eu acho que não há dúvida de que Neville não poderia ter pego isso.

A citação crucial, para mim, segue da citação que a Adamant fornece:

'Then - it might not be me?' said Harry.

'I am afraid,' said Dumbledore slowly, looking as though every word cost him a great effort, 'that there is no doubt that it is you.'

'But you said - Neville was born at the end of July, too - and his mum and dad -'

'You are forgetting the next part of the prophecy, the final identifying feature of the boy who could vanquish Voldemort ... Voldemort himself would mark him as his equal. And so he did, Harry. He chose you, not Neville. He gave you the scar that has proved both blessing and curse.'

'But he might have chosen wrong!' said Harry. 'He might have marked the wrong person!'

'He chose the boy he thought most likely to be a danger to him.' said Dumbledore. 'And notice this, Harry: he chose, not the pure-blood (which, according to his creed, is the only kind of wizard worth being or knowing) but the half-blood, like himself. He saw himself in you before he had ever seen you, and in marking you with that scar, he did not kill you, as he intended, but gave you powers, and a future, which have fitted you to escape him not once, but four times so far - something that neither your parents, nor Neville's parents, ever achieved.'

Harry Potter and the Order of the Phoenix - p.742 - Bloomsbury - Chapter 37, The Lost Prophecy

A Profecia é bastante explícita. ¹

the Dark Lord will mark him as his equal,

Este era Harry, não Neville.

but he will have power the Dark Lord knows not

'You are protetcted [from the desire to become one of Voldemort's followers], in short, by your ability to love!' said Dumbledore loudly. 'The only protection that can work against the lure of power like Voldemort's! In spite of the temptation you have endured, all the suffering, you remain pure of heart, just as pure as you were at the age of eleven, when you stared into a mirror that reflected your heart's desire, and it showed you only the way to thwart Lord Voldemort, and not immortality or riches. Harry, have you any idea how few wizards could have seen what you saw in that mirror?

Harry Potter and the Half-Blood Prince - p.478 - Bloomsbury - Chapter 23, Horcruxes

Neville teria demonstrado tal amor? A mãe de Neville teria morrido por ele e lhe dado uma proteção inquebrável? Possivelmente. Mas isso não importa. Porque ela não fez, ela fez? Isso não aconteceu com Neville, não foi? A Profecia fala de alguém que terá um poder que o Lorde das Trevas não conhece, e uma parte disso parece ser o charme de sua mãe, e Neville não tem isso, se ele poderia teve, ou teria tido é imaterial, ele não não tem.

E em qualquer caso, o Lorde das Trevas irá marcá-lo como seu igual . A Profecia prevê Voldemort escolhendo um menino, e esse menino terá o poder de derrotá-lo. Voldemort pega um menino e dá ao menino o poder de derrotá-lo. Esse menino não é Neville.

¹ Harry Potter e a Ordem da Fênix - p.741 - Bloomsbury - Capítulo 37, A Profecia Perdida

Embora não seja 100% confiável, o HP Wiki nos diz que os Inomináveis fizeram a mudar para a tag depois do ataque de Voldemort.