The best time to go depends on the tour you book: France, Spain, Portugal, Italy is at best in Spring and Autumn (fall). For the Canaries and the Balearics, winter is also very good. Ireland and UK are best in Summer, but sometimes Spring and Autumn are better than Summer.
The weather is great for cycling, with fresh mornings and evenings, and warm and dry days with an occasional brief shower later in the day or evening. You can expect good weather for cycling.