It's very tempting to think that on a practical level, people 'need to know' what's going wrong. Unfortunately it is often very difficult to accurately or objectively assess what the underlying problems are and in many cases, what you perceive as a problem can actually turn out to be a symptom of some other issue.
Project slippage happens all the time - and in and of itself it may not indicate any serious issues. It could just be a case of poor estimation, or scope creep, or some other 'process' related issue which is unlikely to be something any single individual can do anything about.
In my experience, the best way to deal with these kinds of situations is to make sure you're keeping your relevant colleagues up to date. If you feel like some deadline isn't going to be hit - it's probably best to voice your concern about that risk (and only that risk) to your immediate superior. In many cases it's likely that people are well aware of how behind things are - but everyone has a different assessment as to why.
In such situations it's often the case that the most you can do is to tend your own garden. It's important to keep the chain of command - even in small organisations (and especially when dealing with multiple teams / stakeholders), and avoid trying to assign blame (or even identify 'the cause') unless specifically tasked with figuring it out.
Basically: if it's not your job to raise hell about things, then don't. Keep your superiors up to date with progress and let them figure out the implications.
Although it can be intensely boring, project management tools can help to de-politicise these kinds of situations: graphs aren't personal, and the more regularly the relevant people can get a snapshot of progress, the less likely it is that it'll fall to you to point out when things aren't going as expected.
if they are your people then there are all sorts of opportunities to reorganize what gets done first, and who is working on what, and how it all fits together.
you can take a unfocussed seemingly infinite amount of work and turn it into 'if we can just get this seed part done, then the rest can hang off that. jane and timmy, since you are both working adjacent to that piece, and I trust you both to sit down and make it happen - please do that'
thats the best part of the job