Hello, zheniC
Based on your post, it sounds like the transfer of data using Quick Start is taking longer than anticipated.
If you have a lot of pictures and apps, etc this can take some time. During that time, it's best to make sure you stay connected to Wi-Fi and if you're able to keep your device plugged in it will allow your pictures to not pause the download to save battery.
Also, ensure that your device can maintain a consistent and strong internet connection while it is attempting to transfer data to the iPhone 14.
If you're seeing any error messages as things are loading or if you're not seeing any progress being made, such as all apps are showing as 'Waiting..' then restarting your iPhone can also help. Restart your iPhone
Let us know more about what you're seeing and we'll provide any additional details we can.
Take care