Metrics and Analytics
Have you ever had a project where for six months, everyone said that everything is on schedule, then two weeks before the delivery date, someone said there will be a two month slip? If so, then you probably aren't running a data-driven business. When you monitor the right data, you don't need to ask for status. You can see it. Any time. You need transparency and accountability.
Transparency
Your people need to understand that you are not saying "You need to work extra hard". You need to say "I want you to plan something achievable, then I want you to achieve it. Then I want you to repeat it. No last-minute heroics or firefighting. No theatrics or smoke and mirrors with the metrics. Have the experts estimate and plan it, then execute it." Then you need to actually put your money where your mouth is, and back up their plan, even if that means telling the customer it will be delivered two months later than they want it. Or, a better approach is to have your team estimate several scenarios, and let the customer choose one. "You can have features X and Y by August, or you can have features X, Y, and Z by October." (Besides, you know what is going to happen if you tell the customer that you will deliver X, Y, and Z by August - it will take until October anyway.)
The project status should be transparent. It should be posted where it will be seen by everyone working on the project. If the project is behind schedule, will they work harder to get on track? Not if they think the schedule is impossible in the first place. You don't need to harp on your people. If everyone has estimated their own work, they will work extra hard to prove their own estimates.
Project status is not just a stoplight. A single stoplight isn't useful. If it's yellow, did it just become yellow from red or green? Is it trending up or down? Is it a problem at all? The project status is a set of metrics based on the project goals of cost, schedule, quality, and scope. Are we on time? Is each critical component of the project on time? Have we met the quality goals that we set for this point in time? Have we met our cost goals?
Managers will hem and haw about the data. "We don't have that data" or "We know the status of the project" or "It's complicated" or any number of excuses. This is because they don't want to provide the data because then you will have transparency and know the truth.
Transparency includes communicating information to everyone on the project. Sometimes an end date comes and goes and some people are not informed of the project status. Everyone on the project must be informed that the project was released or not released, and why. This builds trust and the people who have less work might volunteer to help fix the problems.
Accountability
Now that you have transparency, you need to hold your people accountable to their goals. Corporate goals (you have some, don't you?) should trickle down into department goals which lead to project goals. Everyone should have some ties to the goals in their annual performance plans. In fact, you can just call them "accountability plans".
People at lower levels tend to feel that their work is not related to the corporate goals. They say "My job is to work hard and get my project done on time, what else am I supposed to do?" That statement might be true - but it is still related to the corporate goals. If a corporate goal is to improve revenue by $10 million, then every department or product should have some specific slice of that $10 million they are going to contribute, and that should all be transparent to everyone in the company. If that is planned as it should be, then you can say "Bob, if we finish the project on time, we will make $X million; if your feature is late, we might only make $Y million." Of course, Bob only truly has a stake in it if he contributed the estimates and believes the plan to be achievable.
Of course, what good are goals without rewards? If the only reward is keeping your job, then that will only work until the job market improves. People who achieve their goals must be rewarded in some meaningful way.