It doesn't sound like it "squeaks" quite at the rate that would make me think it's a roller, water/ps pump, etc.. Like perhaps a damaged spot on the serpentine or timing belt? I'd check the serpentine belt first. Consider that after sitting overnight, running the glow plugs and starter motor, the alternator is working harder for the first few minutes to replenish that power. The extra strain on the serpentine belt could manifest in a squeak like that. It would be easy to pop that belt off and see if the squeak goes away.
Probably ought' to check the timing belt too.