They claim TOS Spock went into the the Abramsverse.
No, he didn't. We don't know where he went, if he even went anywhere.
So, to clarify:
TOS Spock goes in a black hole and then a different Spock comes out of the black hole into a new universe that had been made into a nunew universe because Nero had gone into the same Black Hole that TOS Spock went into and came out 25 years earlier than Magic Spock (the Spock that magically appeared in place of TOS Spock) and messed around with timeline of the new universe, thus creating the nunew universe. For some reason, the same Nero that went into the black hole came out the other end, unlike TOS Spock who was replaced by Magic Spock. TOS Spock is MIA and Magic Spock meets nuKirk who tells him about Magic Kirk (the Kirk that Magic Spock knew in his magic future). nuKirk and nuSpock defeat Original Nero and Magic Spock meets nuSpock at the end.