How much time should someone take to respond to a PhD admission offer?

As soon as you are sure, or April 15, whichever is sooner.

You have no ethical obligation to answer before the April 15 deadline, especially if you are waiting for an offer from another department.

On the other hand, it would be nice to answer earlier if you can. So it would be nice of you to contact your current department's admissions committee (either directly or through your lab director), tell them that you have another admission offer but would prefer them, and ask if they're likely to offer you admission. If they haven't decided, it would be nice of you tell them about your April 15 deadline and ask them when they expect to make up their mind.

(I'm assuming that you prefer to stay in your current lab. If you'd rather accept the pending offer, even if you got an offer from your current lab, then what are you waiting for? If you're not sure, then what more information do you need?)

In Europe there is no 15th of April deadline and nobody cares about US deadlines (if anything the top institutions would want to put pressure on people to make decisions quick so that they don't loose them as good PhD students are not that many (yes its sad but true)). There are many early cycles e.g. Oxford and Cambridge give people offers before the application deadline date for many US departments. And lets say if you get a funded offer in November if your supervisor is keen you can arrange to start by Trinity term in April.

Also this can be the case if the offer was made in irregular time of the year which is not that uncommon (e.g. in Sweden you can start Phd anytime during the year as soon as the guaranteed funding is available for the entirety of the PhD).

In many places you will be interviewed and asked whether you will accept the offer or not just to make sure you are not going to waste their time for a long time. You might say this is unethical behavior and I agree but it has happened to myself a couple of times. You will be expected to make a decision within reasonable amount of time and that is specific to the institution and how they run their business. Back in the day I had offers that gave me less than a month or ones that were open for a couple of months. If the position is funded and there is an expectation to start ASAP especially if there is a company behind the project you might be called in and be given an ultimatum and I have seen offers being retracted.

Advise: talk with the institution that has given you the offer, explain the situation, and get a date by which you can reply. If it is close enough you might be able to stretch it a bit by asking politely for a week or two more time.