This setup isn't too bad (there are plenty of video games out there with less realistic biome configurations than this), but there are still a few things I find odd.
1. The placement of your city/cities
As I understand it, the light blue biome is all one city, and the red biome is either a second city or an extension of the light blue city. However, looking at the surrounding biomes (mountains on two sides, jungles and forests on the other two), this doesn't seem like a very logical place for such a large city to have developed. Especially when there's a nice coastal area only a dozen miles to the west, that seems much more suitable. I'd move your city over there, next to the ocean, and add a large port and/or dockyard on the coastline.
2. A jungle next to a desert
Your intuition is correct here. These biomes are too close, and it would break my immersion as a gamer to go straight from a jungle into a desert. You're better off placing your desert near your mountains; mountains act as a rain shield, stopping water from reaching the desert and leading to its arid climate. You'll also want at least one river flowing from your mountains to the sea (preferably, through your city).
3. The nuclear wasteland
This is right in the middle of your mountain range. It seems like a very odd target for a nuclear strike. I don't see why this particular area would have been nuked, and not the large city/cities only a few miles away. At the very least, you'll want to explain it within the game's backstory.
4. Don't make the coast a straight line
I feel like this is an obvious one, and that the boundaries on your map are just rough examples that will be made more precise when you create the full version for the game, but I might as well note it anyway. Real coastlines curve and meander all over the place, and aren't ten-mile-long straight lines like you've drawn here. This question has a few tips for creating realistic coastlines, that may come in handy for you.