Your mobile device is connected to the RC wifi? That means no internet connectivity. My GO4 never checks for updates without being connected to the Spark or RC, but when I am connected to one of them the firmware check completes instantly because there is no internet.

The Spark can be updated with Assitant but the remote cannot. How will this work?

Hi,
DJI has a video to show how to update RC firmware:

for iOS there is a feature called wifi assist.. this helps

https://support.apple.com/en-us/HT205296