Regardless of your soil, using hard water longterm can cause problems. One of the most visible is the buildup of limestone deposits. You get them around the top of the pot, on the plant if you aren't careful, and out of sight they can clog up the soil. The deposits can be dissolved away using an acid but this is a slow process, perhaps best done by leaving the pot out in the rain for a few weeks (you'll goodness Britain sends all that acid rain!).
Hard water on an acid compost is a particularly bad combination. The acid soil, mainly the organic components, and the alkaline water will react and the result tends to be a nasty mess.
Hard water on an alkaline soil shouldn't cause any immediate problems, but may produce nutrient deficiencies. Certain nutrients are not available in alkaline conditions and one of the ways that plants get access to them is because slightly acid rainwater makes them available. There are other mechanisms involving acidic secretions from the roots and symbiosis with fungi and bacteria, but just remember nutrient deficiency of the plants start to look a big jaded. One nutrient that is always in short supply on alkaline soils is nitrogen, and limestone specialists are usually slow growing plants.