You have an excellent POD there, more realistic than "the soviets land on the Moon first".
The truth is, the soviets were actually the first to shoot LIVING BEINGS around the Moon, in September 1968. But turtles and worms didn't fared well against Frank Borman or Jim Lovell.
A priori, on the
short term in doesn't change a lot of thing; sooner or later Apollo 11 will make Zond pathetic, and the Soviets can't land on the Moon before 1972 at best (or even later, since the last N-1 still exploded in November of that year... 1974 may be more realistic).
As for the
long term - who knows. It may keep the lunar race alive, since in this ATL the Soviets were first around the Moon. That's an event that may have serious consequences.
For example, late 1968 future President Nixon hired the 1964 Nobel Prize, laser inventor and space fan Charles H. Townes as its space advisor. In January 1969 Townes report recommended Nixon to continue Apollo. Townes, however, was not NASA, and NASA had its word to say, and administrator Tom Paine asked for Mars instead.
Nixon answer was a strong "no" and we ended with the shuttle. Sigh.
Now if the Soviets are the first to turn a loop around the Moon late 1968, then Townes may be heard by Nixon... and Apollo may not stop. See my signature: I butterflied away Tom Paine, which is arguably the worse administrator NASA ever had in its history (yes, worse than Mike Griffin himself).