There isn't going to be a next time for Nixon if he doesn't walk away from '68 with the presidency. It's already enough of a historical aberration to nominate a former loser (though less so than today), the Republicans certainly aren't going to nominate Nixon again in '72.
I think there are three plausible scenarios, which I've ordered in terms of how likely I think they are:
1. Some sort of deal is cut, either in the Electoral College or the House of Representatives, between the Democrats and Republicans to put a caretaker administration in power for 4 years. The most likely outcome here is President Humphrey, but with some Republicans in his cabinet and promising not to expand the War on Poverty.
2. Nixon makes a deal with Wallace. I don't see this one as particularly likely, just bc Nixon has to know that if he does a second corrupt bargain for the presidency with George Wallace of all people, his presidency is DOA. I think even someone as singular minded about taking the presidency as Nixon would balk at that offer.
3. Things are hopelessly deadlocked, and everyone just sort of gives up and lets Acting President Muskie be president for his 4 years. I don't think this is likely, just because I think people would expect some sort of resolution to this mess at some point, and just letting Muskie play act as president bc 51 Senators voted for him doesn't feel very satisfactory.
In any of these scenarios, I see a Hart-Celler Amendment having a real shot. One of the primary objections to it OTL was that there wasn't a need for it, that the Electoral College had worked fine so far and there was no need to change it now. That obviously couldn't be further from the truth here, and in fact you now have a very credible real life case for why it needs to be abolished. If it can get through the Senate, which I think it can with more public pressure for it, it's as good as passed. Even OTL, there were 30 state legislatures ready to ratify and a further 6 on the fence. With such a huge crisis, I think those 6 would go over, and then you only need 2 more. In Scenarios 1 and 3, either President Humphrey or (Acting) President Muskie probably support the bill, and in Scenario 2 Nixon has lost enough credibility that I don't think it matters what he says.