How long does it take for a major payment to influence better terms on credit cards or personal loans?

Whatever impact your payment will have, it will have it when your card provider(s) report your balances to the credit bureaus. That will be within a month of when you make the payment but it may be less depending on what point in the month you make your payment and your card provider(s) report.

While the amount of outstanding balances has an impact on the credit score, it isn't going to be a large impact for most people. And most credit decisions are based on the bucket your credit score falls into not the actual score. If the company considers 800-850 excellent, 740-799 very good, and 670-739 good (common ranges for FICO scores), it doesn't matter much if your score goes from 747 to 767 (both very good scores). But it would matter if your score went from 737 to 747 (good to very good). Unless your current score happens to be very close to a cutoff point and/or you are using an exceptionally large amount of your available credit, your payment won't have much meaningful impact on your credit score.

It may have an impact on the size of the credit line you'll be given but that will depend on how much credit you've been given and what sort of risk you are. A 10k payoff probably increases the amount of your credit line by a few k. The payoff makes you less risky because you're using less of your available credit but you still have all that available credit waiting to be tapped which bumps your riskiness back up. Of course, if you have a tiny amount of available credit for someone with your income, a large payment might have a large impact on the credit line you'd get approved for. If you already have credit lines that exceed what you could reasonably repay, a large payment probably won't have any impact on the line you'd get approved for (credit card companies expect that a lot of people that get a balance transfer are going to end up running up their balances again and will not be able to pay off the transferred balance before the 0% offer expires).

In summary, the payment probably isn't going to have a huge influence on what you get offered. Whether it actually will, though, will depend on a number of attributes of your particular financial situation. It'll be reflected soon enough that you might as well wait for the payment to be reported barring an emergency.