It wouldn't be too hard to have the Soviets first
around the Moon early 1969.
Let's suppose the second Saturn V, April 4, 1968, blew up in flight. OTL the flight was a disaster, none of the three stages working too well (S-IC pogoed, S-II lost engines, S-IVB did not restarted).
George Low decision of sending Apollo 8 around the Moon without a LM would be even harder, if not impossible.
In fact Apollo 8 might be a hybrid of Apollo 7 and Apollo 9 - low Earth orbit with a Saturn V but without a LM.
First Apollo to orbit the Moon would be Apollo 10 in May 1969.
Meanwhile on the other side of the Iron Curtain, Zond 7 was to carry Leonov and Makarov around the Moon in March 1969.
http://www.friends-partners.org/mwade/craft/soyz7kl1.htm
I wonder if the Soviet first around the Moon could turn the tide of the Moon race...