In simple explanation, we won't able to know what's up with New Horizons right away. The probe will be busy collecting all the science data during the high-speed flyby.
Closest approach will be at a distance of 3,500 kilometers at about 05:33 on 1 January UTC, and it’ll happen at a zippy 14.16 kilometers per second. 
As happened at Pluto, New Horizons will not be communicating with Earth during closest approach, because it will be focused on gathering all the science it can during the high-speed flyby. Whenever it does turn back to point at Earth to transmit data it will take more than 6 hours for its data to traverse the distance between us. 
However, there will still be downlinks from New Horizons around its January 1 flyby available but don't expect it to be either good or punctual1:
Credit: ESA/Rosetta/MPS for OSIRIS Team MPS/UPD/LAM/IAA/SSO/INTA/UPM/DASP/IDA/Emily Lakdawalla from this page1
WHAT KINDS OF IMAGES OF 2014 MU69 WILL BE AVAILABLE WITHIN DAYS OF THE FLYBY?
The downlinks from New Horizons around its 1 January 2019 flyby of 2014 MU69 will not contain its highest-resolution images. Instead, they will be photos that the team is reasonably confident will contain an image of 2014 MU69.1 Two "failsafe" downlinks are planned for before the closest approach, and three "New York Times" downlinks are planned over the two days after closest approach. These are Rosetta OSIRIS images of comet 67P/Churyumov-Gerasimenko, scaled to be approximately the same size that the New Horizons images of 67P are expected to be, with some processing to add blur and speckle noise (for a variety of reasons, New Horizons LORRI images of MU69 will not look as crisp as Rosetta OSIRIS images of 67P). They are at a phase angle of 10 degrees, similar to the 11-degree phase at which New Horizons will see MU69 from a distance.
One important caveat: the times reported above are when the images will be downlinked. This is not the same as when they will be published. New Horizons (unlike Curiosity, Opportunity, InSight, solar missions, and formerly Cassini) doesn't push images straight to the Web once they land on Earth. The mission will process them, and the team will write captions, and then NASA will have to vet the captions, and then NASA will publish the images at a time of day that'll maximize news coverage, all of which means it could be up to a day or so after downlink that these images get released.
1. A flyby happened and New Horizons did or may not have survived it
Currently, we are in the Approach phase, which began on 16 August and runs through 24 December (7 days before closest approach). During approach, New Horizons has been taking images for optical navigation and searching for potentially dust-generating hazards like rings or moons. (So far, it has not spotted any.) The in-situ instruments have been busy gathering data on the fields and particles in interplanetary space. 
Furthermore, the operations team weighed numerous factors in making its choice to closest approach. The considerations included what is known about MU69’s size, shape, and the likelihood of hazards near it, the challenges of navigating close to MU69 while obtaining sharp and well-exposed images, and other spacecraft resources and capabilities.2 It's unlikely that New Horizons will not survive the flyby.
2. A photo or spectrum or some measurement of MU69 itself
As said above, New Horizons has been snapping pictures of Ultima Thule to identify any potential hazards. The result so far is that it appears clear and safe to approach. Earliest downlinks will be available on January 1, the rest will released accordingly during the Departure phase that will last one week after the 2-day period around closest approach, from 3 through 8 January. After that, New Horizons will spend 20 months downlinking all the remaining data, until September 2020.
Regardless of U.S. government shutdown, the show will go on!
New Horizon flyby of Ultima Thule event timeline (from this page1):
Source: Partial screenshot of this page1
1 Emily Lakdawalla, December 17, 2018, What to Expect When New Horizons Visits 2014 MU69, Ultima Thule (And When We Will Get Pictures), The Planetary Society.
2 Johns Hopkins University Applied Physics Laboratory (2017), 6 September 2017, New Horizons Files Flight Plan for 2019 Flyby.
Quoting New Horizons Principal Investigator Alan Stern of the Southwest Research Institute (SwRI), Boulder, Colorado.
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