Lots of issues in one post.

First- 7 speed freewheels (or cassettes, there's a difference that you need to be sure of) are a bit wider then 6 speed ones. It only takes a few mm's more cog width to have the chain contact the frame/dropout. While a axle spacer could be added to get the RH end of the axle to stick out further then the 7 speed FW does it's not like more gears will make you a better rider. So the easiest is to stick with what works already, the 6 speed cog set.

The fancy wheels likely have a longer axle, the 6 speed might be a 126mm long axle lock nut out side to locknut out side, the fancy ones are likely a 130mm axle width. So some frame spreading is needed to install the fancy wheels.

All of this wheel and cog count refitting is over laid on the possibility of the bike having an indexed gear system. this means that without a friction shifter option the cog count needs to remain the same as the shifter or problems will occur.

I suggest that the Op does some more homework before changing anything. Andy.