Most of the Wasatch areas with good backpacking are higher up. It's really just a crapshoot in late October. Could be snow-free and not frigid, but it could also be snowy and very cold.
I'm inclined to say that you'll have decent luck with snow because October has been fairly mild over the past few years, but the mountains already got a nice covering over the weekend, with almost a foot up in the Uintas, so I'm doubting any intuition I've gained in 6 years of living here. I was in the Uintas in the second week of October last year, and it was fine. Other users here went even later and it was good for them. But that has little bearing on this year.
So, like Nick said, it'll be very good to have a backup plan south of Salt Lake City and Provo.
But you asked for recommendations, so I'll offer a few. Most people will not be backpacking then, so you can get in and see some of the more popular areas and have them all to yourself. Try Naturalist Basin or Amethyst Basin in the Uintas. Plenty of exploring to do once you get a basecamp set up.
For the Wasatch, I am not sure about trips that are more than an overnighter. You could hike in and camp in the meadows below Timpanogos, summit the next day, and spend the night in camp again. Or you could do two or three overnighters, in which case I'd recommend Lake Blanche, Red Pine Lake, and maybe a summit of one of the higher peaks along the Front. Both of those lakes are incredibly popular in the Summer, by virtue of being very close to Salt Lake City, so late October would be a good time to head up (conditions permitting).