Whether you are the captain of your own ship or just an officer on the deck, there's a lot you can do as game producer to improve timeliness and quality. Here's a checklist out of the Game Producer's bag of tricks.
Scope Check Early
If you are an agile shop or still doing waterfall scheduling, you should do a scope check on the design as early as possible. Use ballpark estimates or WAG (wild-ass-guesses) in man months or SCRUM story points on a per feature and per resource type basis and level it to your resource headcount or velocity. Make heavy use of Excel. If this isn't your first time around the block or it's not completely original, then you should have historical data you can pull from. Pad anything that's completely new or in other ways fraught with risk. Push back on the design or get more time or resources using these hard-to-dispute numbers. Get other people's estimates to back your own if necessary.
Trust Your Gut and Fact Check
This goes for anyone, but especially producers - you should not repeat bad information. Verify it first. The time saved in clearing up confusing or inaccurate information is enormous. Plus you never want to be wrong when you communicate up the chain. If your gut tells you something might not be true, verify it personally or using whatever tools and people are at your disposal.
You are the grease in the wheels and the plumber's snake in the pipe. This doesn't mean micromanage - just be in the loop and nudge things over the hurdles. More than anyone else, you should have the widest view of the status of your project and be able to repeat it on demand. If there's a meeting about a problem, you are the one who calls the meeting and gets the problem solvers in attendance; you're not the one who solves it.
Embed Testers and Test Early
An embedding testing team and testing well before Alpha can help you generate a more accurate picture of the status of your project and a better chance of hitting your milestone. Testing the day or week of a milestone leaves very little time to actually fix the bugs or do any sort of polish.
Improve your Bug Test Matrix
Your embedded testers are an extension of your eyes and ears. They can help you, but don't rely on your testers exclusively. It's ultimately going to hurt you more if they missed something. Do your own tests. Hold your testers feet to the fire if you're finding bugs they're not, then improve and check their test plans. The better net you cast in embedded testing, the faster you'll get through formal testing.
Discipline Lead Sign-Offs
Don't trust quality to your lowest paid people. Your leads should be constantly playing your game. By simply adding a lead sign-off requirement before hand-off to formal testing or deployment, you will see your leads rise to the occasion and take ownership where it needs to be, in the hands of the discipline experts. Your sign off is the hand off. Don't hand it off until you got all the lead sign offs or their concerns entered as bugs.
Own Your Mistakes and Wear your Issues like a Badge
You cannot get better if you don't acknowledge mistakes and you cannot be completely held responsible if they can't say you didn't warn them. This is a people operation, so you are never in complete control. Broadcast when something is slowing you down, but also include your suggested resolution. Your directors should not have to solve all your problems for you, but they should be kept in the loop. If they don't hear about a 2 week slip until the original due date, then you're doing something wrong.
Manage Wide and Up
Your job is to help your boss succeed with his or her goals. Your co-workers' jobs should also be the same. Yet sometimes your co-workers and even your boss are not aware that some action on their part is severely impacting the schedule or quality of the game. Don't be afraid to point this out. You should not be a yes-man, but you need not be a total gate-keeper either. State your case. Be prepared to explain your reasoning and the impact. Be prepared to step back and accept their response. You're not the best judge if the issue is outside your expertise. They might surprise you and agree or come up with an alternative solution that alleviates your concern.
Keep Your Eyes Focused on the End Goal
Of all the people on the project, the producer needs to keep a look out for down stream impact of decisions made today that may risk project milestones and long term plans. Keep an eye on planned vacations and subsequent resource shortages. Keep an eye on the creeping bug count and plan for time to fix them. Don't let feature creep, excessive iterations or nitpicking push you past your milestone dates. Prioritize your bugs. If you're the scrum master (and you should be!), police your backlog. Make sure your polish is all fine strokes, not painful broad stroke changes. If your scrum team members are rolling too much over, call them on it and improve your planning. If your team members are coasting, push more onto their plate. If a task is swirling, find out why and fix or reassign it. More than anyone else, as producer you uphold the timeline goals.