The first thing you gotta check is the hub spacing. Older frames were 120mm, newer frames are like 130 mm to accomodate the extra gears. So, your newer wheelset might not fit the older frame. Sometimes you can spread the rear triangle by force enough to cram the rear wheel in, but I wouldn't recommend it. Maybe 3 or 4 mm, but not 10.
Then, your 6 speed derailler won't shift an 8 or 9 speed chain with consistency - especially if you intend to do index shifting. Chain is much narrower, and the distance from top jockey pulley to cogs is closer. All this stuff is engineered by the manufacturer to interface just so in order for the index shifting to work right. You probably know you have to be careful when upgrading - even with the same manufacturer. 15 years, plus 6 to 9 speed, plus who knows mis-matched manufacturer. More trouble than it's worth.
I've seen it done. It ain't easy, but people with bikes that have deep sentimental value go to unbelievable ends to keep it going.
Why not go to 7 speed? You should be able to squeeze an extra gear in there. And, since you don't like Biopace, install a new triple crank - with carefully chosen rings! This will help minimize gearing overlap, and 21 speeds is plenty. My Schwinn Hybrid is 7 speed freewheel (13-21), Huret front and rear deraillers, and the crank has hand-picked rings 46(alloy), 36(steel). I've never needed more gears, and I've had it 5 years now.