I have a mulberry tree which has been in my garden for 4 years. I bought it at a market and I think it was a seedling. It has grown to a height of 4 metres and is thriving. Each year it has flowered and the bees are in abundance. After it has flowered the flower part shrivels, goes brown and drops off. I have not had one single mulberry from it. It has been fertilized with organic fertilizer and is always watered. Our climate is ideal for mulberries. Our neighbour's tree produces heaps of fruit.
One possibility is that you have been sold a male tree which also flowers but doesn't bear fruit.
Commercial varieties sold from nurseries are typically self pollinating so you only need a single tree (although you should check the label). However, in the wild you will see male and female trees and only the female bearing fruit.
UPDATE: I just thought, one option you could consider is grafting a branch or two from your neighbours tree onto your tree and you should be right to go.
I won't go into details on how to do this as there are thousands of articles online that describe how to do this.
My suggestion is to maintain one or two strong branches from your existing tree and then graft on a couple from the fruiting tree at a stable position.