In short, Nixon probably still becomes President if he reaches a deal with Wallace before the Electoral College assembles, but Humphrey is more likely to win if it reaches the House of Representatives without a deal having been reached.
Let's put aside the how of there being no majority in the Electoral College and focus on the outcome. Likewise, let's look at it methodically. The possible outcomes are: Nixon throws it to Humphrey, Humphrey throws it to Nixon, Wallace throws it to Nixon, Wallace throws it to Humphrey, or an Acting President deadlock.
There would first be a negotiation phase before the Electoral College has assembled. Wallace was pretty vague on what he actually wanted out of a deadlocked election besides slowing or reversing desegregation. Practically speaking, Wallace had to know that neither candidate is going to reverse desegregation, so it comes down to the extent that either is willing to slow things down. Humphrey is too moralistic and too tied to civil rights to negotiate with Wallace, so that's a non-starter. While Nixon was supportive of civil rights in principle, he wasn't tied to it like Humphrey, and would be more willing to negotiate on it, and Wallace's "Law and Order" culture war inclinations are much closer to Nixon than Humphrey. Likewise, by 1968, Nixon was basically obsessed with becoming President, so he wouldn't be willing to throw it to Humphrey. That means, in direct negotiations, either Wallace would throw it to Nixon or Humphrey would throw it Nixon to block Wallace from having any influence, but it would probably be in Humphrey's best interest to wait it out until it goes to the House of Representatives.
Assuming it goes to the House, the Democrats theoretically have a 26-to-24 majority that could win it for Humphrey immediately, but the Southern Democrats would likely attempt to extract concessions. Removing the Southern states, Nixon has 19, Humphrey has 14, and 3 are mixed delegations, so the South does indeed have swing power to determine the election in the House. However, Southern Democratic delegations would be more willing to work with Humphrey due to party loyalty and Southern political dynamics, so it would be in Nixon's best interest to settle things before the Electoral College has assembled.
Ultimately, this means that the most likely outcome is Wallace and Nixon reaching an agreement before the Electoral College assembles, and Wallace instructs his electors to vote for Nixon. Now, if Nixon wants to put his crookedness to good use for a change, he could pursue the same quiet, behind-the-scenes integration that he did IOTL and hope that Wallace doesn't notice. Alternatively, if Nixon actually follows through with the deal, then perhaps he reverses the Department of Health, Education, and Welfare's ruling that the South had to implement busing desegregation.