I have never worked with cucumber cuttings but I have with melons. Both of these plants belong to the same genus, Cucumis, so I'm going to assume that they react very similarly. I took a cutting from one of my heirloom melons so that I could root it far from other melons to avoid cross-pollination. I thought it would be best to place the cutting in lightly fertilized water so that it could root. However, it initially only formed little stubs that looked like roots at the first node. It didn't get past this stage as long as I had it in the water. Soon after I moved it to a pot with potting mix, the roots took off growing.

I wish I could answer your question with certainty but I don't remember how many days I had it in the water. If I had to guess I would say about 10 days in the water and about 4 days to start growing once I put it in the pot with potting mix. My effort was ultimately futile because the poor thing developed powdery mildew.