You called?
I no longer have access to the period Newsweeks which had the weekly delegate tracker, but IIRC as of June 3 HHH was ahead by 250-odd delegates, but still 200 short of the 1314 needed for nomination. Something like 1150-858. Remember that these numbers are not hard: 75% of the delegates are boss-controlled, with only 25% chosen in primaries. Daley and Dan Rostenkowski strongly hinted that if RFK won California they would support him.
If Kennedy lives (very easy: the bullet entering one inch higher means it lodges in the back of his skull instead of splintering into his brain as per OTL- had he not died IOTL he would've been a Schiavoesque vegetable), then he'll have won California 46-42 (the only WTA primary, so he gets 174 delegates) and HHH's home state of South Dakota 50-30-20. On June 11 he has to win NY by a convincing margin since McCarthy was mounting a strong challenge. June 18th is the preferential Illinois primary, which is nonbinding and RFK had avoided entering at Daley's request. After that it's 2 months till the Democratic convention.
To win Kennedy has to convince the Democratic bosses that the primaries proved he is the best candidate to hold the NDC together in the fall. Which is true: his core voters are blue-collar workers (many of whom defected to Wallace or Nixon in November), blacks and Hispanics. Another argument is infrastructure: he had by far the best campaign organization on the Democratic side, not to mention fundraising. If he convinces them, then he can win the nomination on the 3rd or 4th ballot.
How will Gene McCarthy go? Every indication was that they'd support HHH rather than each other, but supporting Humphrey would infuriate McCarthy's antiwar base because of HHH's hawkish stance. Supporting Kennedy is ASB given how much they loathed each other. So McCarthy probably releases his delegates at some point without instructions on who to vote for.
Sympathy vote: He gets a slightly bigger margin in NY, but needed about 5-6 weeks to recuperate from those sort of injuries.
Does he win the nomination: pure tossup.
Can he beat Nixon: again, pure tossup. Wallace's vote will be squeezed outside the South since many Northern RFK voters who voted for Wallae in the general IOTL will stay Democratic. Increased turnout in LA among blacks and Hispanics will keep California in play. It will be a narrow victory either way.
RFK presidency: Overall, slightly to the left of Clinton. You'll see many OTL Clinton initiatives such as welfare reform, AmeriCorps, expanded health coverage for low-income Americans, but also a shot at UHC along similar lines to Nixon's proposed CHIP and higher taxes on the top bracket. FP wise, Vietnamization and diplomatic recognition of the PRC.