You don't need to change the wheel. They all have the same thread, so you can fit any freewheel you want. The standard frame width is 135mm regardless of which freewheel is fitted. When you have more gears on the freewheel, the spacing between the gears can be narrower, so they're not necessarily any wider, but a 7 speed will probably be a bit wider than a 6 speed. You can easily spring the frame that much.
Depending on what changers you have, you might not be able to get all the gears. I have a 6 speed changer.on my Ebay bike, but I can get all seven gears after I upgraded from 6 speed to a seven speed DNP freewheel. It's the cheapest twist type changer - Revoshift?