As the titles says, the shipyard/military overflow seems to be working, but it isn't displaying properly. Following is a series of screen captures from turn to turn showing the display of shipyard production.
This first image shows a Paladin M1 with some production already completed. Two turns prior, I built a Sniper with a manufacturing cost of about 87. I don't know what was built before that, but there was definitely some overflow manufacturing applied.
As you can see, this Paladin costs about 203 and would normally take two full turns of manufacturing. After one turn, however, manufacturing was almost completed.
That shows that the overflow is working, because the overflow from the Sniper nearly doubled the production this turn.
To sum up: turn 1) Sniper built (using some overflow); turn 2) Paladin queued and nearly completed using overflow and normal turn manufacturing.
This next image shows the queue, with the previous Paladin M1 and the next ship, a Paladin M3.
This screen shot was taken on turn 3 of this sequence. You can see that the Paladin M1 is completed as expected, and the Paladin M3 has its full production time remaining (3 turns). This production time makes sense based on the cost (255) and the manufacturing per turn (109). (109+109+37 = 3 turns.)
This final image shows the next turn, turn 4. The previous turn showed a production time of 3 turns for the Paladin M3. After 1 turn, the Paladin M3 has 1 turn remaining. This means that more than 145 manufacturing points were applied to the ship in this turn, despite only having about 109 manufacturing.
So overflow is working (woot!) but not displaying properly on the first turn of overflow.