New Horizons was hidden behind the Sun a few days after the flyby, from January 4th to January 7th.
Almost as soon as the encounter and earliest downlinks are over, New Horizons will go into solar conjunction from January 4 to 7. I asked Alice Bowman what implications that has for the mission, and she replied: "During solar conjunction, the downlink data rates are lower because of interference from the Sun. Generally, because New Horizons has fairly short solar conjunctions, we can stand down from science data playback (as the chances of dropping data is increased); we also make sure not to conduct any critical uplinks or sequence transitions, and we set the spacecraft’s command loss timer to allow it more time to receive commands."
They'll stay in 3-axis mode until after the flyby, returning to spin mode on January 9 to begin dumping data.
in 3-axis mode, the spacecraft attitude is controlled by the thrusters. This mode is used to point the spacecraft instruments at their targets.
In spin mode, the antenna is aimed at Earth and the spacecraft is set to spin on its axis to stabilize it. This allows the antenna to remain aimed in the right direction without using the thrusters.
A paper that summarizes the initial findings based on the first post-encounter download was submitted on 2019-01-09:
At the time of this abstract’s submission, only 4 days of data downlink from the flyby were available; well over an order of magnitude more data will be downlinked by the time of this LPSC meeting in 2019 March. Therefore many additional results not available at the time of this abstract submission will be presented in this review talk.
So they've started downloading on January 9th, for 8-12 hours/day at 1 kbit/s.
The first website update was scheduled for January 11, but that would have given just one day of downlinked data, the next update is scheduled for January 18. Also, LORRI is just one of the 7 instruments on board, and the only one for which raw data is made public within a week.