Today is the day iOS10 finally drops onto iOS devices around the world. Natuarally this means the Apple servers take a beating on the first day as everyone wants to update to the latest version.
I have previously put up a post on how to manually update your iPhone or iPad, which makes the process a whole bunch faster. Check out the step by step guide here -> https://www.marcforrest.com/2011/01/13/how-to-update-your-iphoneipad-firmware-manually/
To download the firmware files directly for iOS10 you can get them here (Thanks to the folks at iClarified)
Latest:
10.0.1 (5 GSM):iPhone_4.0_32bit_10.0.1_14A403_Restore.ipsw
10.0.1 (5 GSM+CDMA): iPhone_4.0_32bit_10.0.1_14A403_Restore.ipsw
10.0.1 (5s GSM): iPhone_4.0_64bit_10.0.1_14A403_Restore.ipsw
10.0.1 (5s GSM+CDMA): iPhone_4.0_64bit_10.0.1_14A403_Restore.ipsw
10.0.1 (5c GSM): iPhone_4.0_32bit_10.0.1_14A403_Restore.ipsw
10.0.1 (5c GSM+CDMA): iPhone_4.0_32bit_10.0.1_14A403_Restore.ipsw
10.0.1 (6+): iPhone_5.5_10.0.1_14A403_Restore.ipsw
10.0.1 (6): iPhone_4.7_10.0.1_14A403_Restore.ipsw
10.0.1 (6s): iPhone_4.7_10.0.1_14A403_Restore.ipsw
10.0.1 (6s+): iPhone_5.5_10.0.1_14A403_Restore.ipsw
10.0.1 (SE): iPhoneSE_10.0.1_14A403_Restore.ipsw
10.0.1 (7): iPhone9,1_10.0.1_14A403_Restore.ipsw
10.0.1 (7+): iPhone9,2_10.0.1_14A403_Restore.ipsw