As for installing a game on either the PS4 or the XB1 you can do this via digital download or by media which in this case is a Bluray disk. For digital downloads the time take is dependent on your internet performance between your console (the same applies to PC's) and the content provider, which for the XB1 is Microsoft. Some players have excelent internet perforace and can download at well over 30 MB/sec and some are lucky if they can download at 500 kB/s. In addition performance can vary dramatically over the course of the day.
With a media install you are only limited to the read speed of the Bluray device and the write performace of your storage disk which is always the limiting factor. Note that for a 5400 rpm hard disk (normally the default for console) write performance is approximatey 20 MB/s. Installing from digital media has the bonus of you not needing to wait very long to actually start playing the game although this does depend on the game. I think some digital download games can let you do something similar but I am not sure.
Either way to completey play the game it must be fully installed and normally this is actualy done in what is called "the background" which means the download continues to install even if you are doing someting else. If you are waiting, well you are waiting.
Many forums discuss the advantages and disadvantages of digital download verse media install. Which one you choose really depends on you keeping in mind that if you remove the game and wish to reinstall via digital download at a later date you may have a long wait for the game to be fully installed again if your internet download capabilty is relatively slow.
A word of caution, some internet providers offer unlimited upload and download, however some will throttle or reduce your speed dependent on the overall network usage. Other providers do have caps in place such that when you exceeed a certain number of Giga Bytes you are charged extra or severly throttled.