I think there's a good chance you'll be able to hike, just not necessarily every trail. Some areas may be closed do to slippery trails or the danger of falling ice and snow from above. That being said, waterfalls and rivers will be raging and the remaining snow will add to the scenery.
Even if it's rainy, snowy and foggy every day, I'd still take a day walking in the Alps over my normal daily routine. You should bring both waterproof jacket and pants, along with synthetic/wool socks, hat and underwear, so that even if it's colder and wet, you'll still be warm and have the chance to stay outside longer. Two pairs of shoes are pretty important items for me as well.
I totally agree that late-June, July and August are the best months to be in the mountains. These months are also the busiest and more crowded. The middle of summer would be my first choice, but even the one trip I took to Switzerland in September that was mostly rainy the whole time was still super fun.