This is not a packet switched network and you are not receiving nice neat little messages to decode, you've got a radio receiver, the real transmissions are from a satellite in space, while unfortunately your adversary is transmitting from radio towers or even a portable transmitter, much closer and able to be supplied with energy easily so more powerful, but on the same frequencies.
Also, be careful with assuming that late = imposter. The whole point of GPS is that signals take time to travel over distance, so, maybe the signal is late because you're further from that satellite instead, which is what you were trying to measure when you got distracted trying to prevent spoofing.
I agree that there is an issue with overwhelming the satellite transmission frequencies. However, the signal can never be delivered earlier by adversary as signal travelling in air is about the physical limit. Perhaps only a stalking satellite could intercept and deliver the message faster, but that seems like a sci-fi.
> got distracted trying to prevent spoofing
Indeed overwhelming the processing power with junk seems the important issue if crypto is added.
Also, be careful with assuming that late = imposter. The whole point of GPS is that signals take time to travel over distance, so, maybe the signal is late because you're further from that satellite instead, which is what you were trying to measure when you got distracted trying to prevent spoofing.