Tug driver reported a ramp safety concern while pushing an aircraft from the gate.

Narrative

I was cleared to push [Gate] XX at ZZZ. We were dropping the plane at the drop spot and [Gate] XY had also pushed into the alley. Causing us to get jet blasted if I continued to bring the tug back the gate. I stop my tug and keep honking at XY crew to stop their push as my crew and I were still in the alley finishing our push. XY finally threw up the X with their wonds. I stopped my tug and waited a couple seconds to see what they were going to do. It was extremely unsafe. Also; the working conditions are extremely slushy; Icey; snowy. Visibility isn't the best either; you cannot see the line on the ramp properly. My tug has been sliding and losing traction throughout the day. They could've had the same problem and would've come in contact with my CC (Crew Chief) and I in the tug. Cause: The event occurred because I was cleared to push from XX during my push XY was cleared as well. Suggestions: Even on a 'good' weather day these approved pushes from both gates at the same time are unsafe. As XX goes a crossed the alley. XY stays on the side that I bring the tug back to the gate.