Firstly, be very careful how you phrase things.
Calling someone incompetent is going to immediately shut down the possibility of actually investigating the root source of the issue. This seem to be built on some unfounded assumption that Management will blindly believe you over T when a conflict becomes apparent.
It is better to assume the exact opposite - you will not be believed until you have concrete evidence to the contrary. Ringing the bell before then is not going to look good on you.
Secondly, treat T and Management as a single entity, because they are both representing the same company.
You have no way of knowing what resource T has, whether the wrong people were hired for this job, whether they are buckling under an excessive workload, or whether they're meeting Management's overall expectation. Don't involve yourself in this.
You've been hired by a company. For all intents and purposes, the company acts as a single blob. This may be a multi-faced blob when you have several points of contact, but you have to treat them as multiple faces of what is essentially a single entity.
The first order of business is agreeing with management who your team depends on, i.e. establish your points of contact, points of escalation, and exactly who is authorized to decide/alter your workload. If someone is not on any of these lists, they don't matter to you and you should not cater to them either.
The rest of this answer assumes you know your proper points of contact.
Formally speaking, there is no purpose to distinguishing that T is not holding up their end of the bargain. T represents the company, and therefore the company is the one not holding up their end. Whether T is the sub-delegate that is supposed to provide the company's end here is up to the company to decide, not you.
Instead, you should communicate with your point of contact, and eventually your point of escalation when the situation does not get resolved. From that point on, it is on the company to fix its own internal issues, you should not meddle in this (unless you are both willing and explicitly asked to).
Is T not providing something they should? Talk to your point of escalation about being blocked because you do not have access to the [agreed upon thing]. Let Management sort out who should be providing [thing] and why they're not doing so. Not your circus, not your monkeys.
Thirdly, stop doing work you were not hired to do.
Failure to provide things you need can shift your deadline and is provably not your fault (make sure to document it to make sure). However, you deciding to work on something that you weren't hired to do, in a way that it impacts your own deadline, is very easily construed to be your fault for choosing to do so after making the agreements. This will bite you in the end if you hit a point of friction with Management about non-delivery.
Instead, alert your point of escalation that you cannot perform the tasks you were hired for. Wait for them to respond to this situation. It may still end up that you need to assist in resolving the issue, but wait for the company to explicitly ask you to do so. At that point, remind them that doing so will impact your deadline as you have now been sidetracked into work that was not originally planned.
If they disagree, do not let yourself get sidetracked, and repeat that the issue needs to be resolved as soon as possible for you to continue working.
If they agree to shift the deadline, that essentially solves the issue that the blocking issue created for you.
Fourthly, you should have factored in these kinds of delays.
You can't account for every possible scenario, but you do mention that you agreed to a tight timeline. That was a mistake on your part.
You effectively agreed to get a handle on a company's internal system, sight unseen, and made promises as to how quickly this could be delivered. You should either have investigated what you were agreeing to, or you should now have agreed to such a strict timeline.
Always pad your estimates with your expected delay times. The amount of padding can vary based on prior experiences with the company (there have been clients where I padded 150% simply because they continually changed their mind and had no concept of locking anything in).
Lastly, document everything.
Every time you are blocked, track the amount of time you are blocked. Back this up with evidence, such as the timestamps on when you sent an email and when you received a reply, or how long it took to resolve an outage, or how much work had to be redone because of someone changing their mind. Use manhours as your unit of measurement. If your entire team is blocked due to an outage, the effective cost is
outage duration * team members.
Don't immediately start blaming the company for the first few delays that happen. However, as you get closer to your deadline or as the delays keep happening frequently, it becomes necessary to alert your point of contact that the deadline is in danger of no longer being reached. At this point, the company will respond about you not reaching the deadline.
That is the point where you bring forth your evidence, and a summarized report of exactly how many manhours have been lost due to blocking issues on the company's part.
Obviously don't just keep the blocking issues a top secret until the last day before the deadline either. Kindly alert people of the blocking issues, but don't immediately start ringing the alarm bell or suggest that this requires a contractual renegotiation, because you will be perceived as hypersensitive and will eventually be thought of as the boy who cried wolf.
- Assume that the company knows precisely what it is doing, and that it has accounted for its own workings.
- Document any and all impact that the company's issues have on your workload. Be precise and provide specific evidence.
- Never butt into the company's private business. You do not have any standing to slander T to another part of the same company.
- Do not waste your time on work that you're not being paid for.
- Always include sufficient padding in your estimates for delays. Never try to undercut your own estimate for the sake of pleasing the client, unless you are making a conscious decision to roll the dice on such an outcome.
- Always make sure the company explicitly agrees to change your focus - and make sure to always mention the impact on your original agreement. This ensures that the company consented to these delays, and you cannot be blamed for going rogue or failing to deliver by mistakes on your own part.