I hope someone can recommend a system that would handle it.
GURPS High Tech, or Burning Chrome maybe? I've never played GURPS, and I haven't played with Thor's "Burning Chrome" hack, which would need to be further hacked to get to Mass Effect from Cyberpunk.
My brother and I have just started playtesting a conversion of 5e to the Mass Effect universe. We've been working on it on and off for the last two years or so. So far, it's been working fairly well. If any of you lot are interested I'm sure I could set uo a game at some point!