Metrically, I'd start breaking each three of the 6/8 into two dotted 8th notes, increasingly emphasizing the new rhythm. You could do this either in the melody first (I'm presuming that to be in the right hand), or you could continue with the melody or chords in the right hand while shifting the left hand's rhythm in this way beneath it.
Eventually, the dotted 8ths will be the 4 quarter notes. You can transition your tempo as needed (speed up or slow down to match the second piece).
During this you can introduce a Bb to your C chord and swell dynamically in anticipation of a key shift to the F major of the second piece. If your segue takes time, you may want to hint toward the F before adding the Bb, to give the listener a taste of leaning F-ward and further set up the shift.