The launch schedule for 1969 was determined in the first days of the year, putting the Apollo 11 launch tentatively on July 15th, although it would depend on the success of Apollo 9 and 10 if Apollo 11 was going to be the landing (G) mission. NASA confirmed the launch date for Apollo 11 only in June 1969 and announced it publicly as the Moon landing mission on July 6th.
The Apollo 11 Press Kit announcing the launch date of July 16th was released on July 6th, 1969, only 10 days before the launch. Internally, the final decision was taken only a few weeks before that:
Armstrong and Aldrin also trained at other places, especially at Langley Research Center, where they worked on the suspended lunar landing trainer equipped with realistic surface views and lighting. On 12 June, NASA senior management agreed that the crew was ready for a 16 July  launch. Less than a month later, on 7 July, Mueller told Paine that "if Apollo 11 continues to progress on plan, the first men will set foot on the moon two weeks from today.
(Chariots For Apollo, chapter 13)
Note that in February 1969 NASA was still building training equipment (the 1/6 gravity simulator).
Nonetheless, internally the earliest possible launch date for G mission was of course known. After the successful Apollo 8 mission, the following launch schedule was determined for 1969:
Frank Borman's Apollo 8 crew in its flight near the moon had met no major obstacles, but the need for trailblazing missions had not lessened. Associate Administrator for Manned Space Flight George Mueller in Washington wrote Center Director Robert Gilruth in Houston after Apollo 8 to remind him, "It is essential that we not rest on our laurels, for we have yet to land on the moon." Gilruth foresaw few chances for resting. Only three days of the new year had passed when John D. Stevenson, Director of Mission Operations in Washington, projected five Apollo flights for 1969, with launches on 28 February, 17 May, 15 July, 12 September, and 10 December. This schedule was essentially the same race-with-the-decade timetable outlined a year earlier.
(Chariots For Apollo, chapter 12)
Apollo 8 launched without an LM, so the E and F missions would still need to follow before the G mission could be attempted. Hence, the first possible opportunity for the G mission was going to be the third launch in 1969, tentatively scheduled on July 15th.
They pushed hard to get Apollo 8 launched in December of 1968. If you look at the operations and launch schedule, you'll see about 2-3 months between launches*. They were working at maximum capacity around that time to push for the end-of-the-decade landing, so assuming they couldn't launch any faster, even optimistically Apollo 11 wasn't going to launch before June 1969.
*) The launch vehicle coincidentally also spends about 2 months on the launch pad before being launched, presumably NASA didn't want to risk (at that time) launching with another stack on the pad, so maybe they wanted to only roll-out the next vehicle after a launch (see also this question and answer). However, given the time pressure at that time I doubt that they could have launched faster if they wanted to take that risk.