What level of physical fitness do I need for the trek?

You need to do few weeks of training prior arriving in Nepal which leads your journey more comfortable. With the full of preparation before trekking in Himalayas would make your journey more rewarding. Try to spend some hours in your Alps; hike to the upstairs, jogging or running (optional) few miles or (15-16 km) every day will probably best physical activity, which will be the more effective when you are in Nepal for trekking.