The TV hadn’t picked up a signal since the great war of 2037, but that didn’t stop me from trying it regularly just to be sure. Yesterday was different. Yesterday something tried to bleed through.
You could use a semicolon instead of a period after different (and not capitalize the second yesterday, of course), but I like the separate sentences because it makes Yesterday was different really stand out, and that sentence seems important here.
Whether or not you capitalize great war depends on whether or not people in the setting are writing history about this event and giving it an official name, etc. I got the impression that the timeline isn’t that far along (and that there wouldn’t be much history writing going on anyway), so I left great war uncapitalized. (This is why authors need to communicate with their editors. Sure, we can guess about such things, but in real life it’s better to be able to ask for and receive clarification.)