"Popping up" the roof in this sort of situation really means adding a whole new story, with new full-height walls all around, and such.
Your house may or may not be framed to handle this added load, so reinforcement (doubled studs) might be needed in the lower floors before you can do this - a MAJOR project, as it means opening up all the house walls above the foundation (at least the long 2 walls) to put in new studs (and rerun plumbing and wiring in the process after new studs are in), so most people would not consider this unless they were planning to replace the siding on the house anyway.
Check your local planning and zoning board - you may have a or height limitation in your area that would prevent this anyway.
Even if you can just remove the existing stub walls (if you have them), the existing upstairs is essentially a finished attic - not a full floor. The original Cape Cod design was done that way to eek some sleeping space (nighttime use only) out of the attic but avoid building an additional story, and to keep the space small to make it easier to heat.
To "raise the roof" will take either a MAJOR roof jacking operation or a crane (I have never heard of this being done on a residential roof) to elevate the roof and hold it there or to lift it in one piece and set it on the ground while new walls are built, or far more likely, tearing off the roof including the sheathing, using a boom truck to individually lower the roof trusses to the ground, building new walls, putting the trusses back up, reroofing, putting on siding, running electric and plumbing, then interior finishing. This is essentially equivalent to building a whole new addition except that you do not need a new foundation.
I seriously doubt if most contractors would get the job done in less than 2 months, which is just what you said you don't want. And - regardless of how long it takes, during that time the upstairs will be non-existent for you - you would have to be living entirely in the downstairs and basement, along with all your upstairs furniture and possessions.
This sort of job is risky - you better have a GOOD contractor - imagine if he gets busy and puts your job on hold for awhile, quits, or tries to raise the price on you mid-job. Also, it is very tough to make this sort of job totally raintight, so you risk water intrusion into the upstairs floor area and into the downstairs ceiling and walls, and your current upstairs floors will most likely be trash too by the time the job is done. Personally, I would consider this type of job only over an unoccupied garage - in your situation I would just build on an addition, which does not greatly disrupt your life.
I would ballpark this type of job roof raising at $50,000-100,000 - are you prepared for that sort of cost to get a relatively small increase in living space ? I really think that if you serious about this, you need to talk to an architect who is experienced in remodels of Cape Cods, or maybe do what most people do when they want more luxurious space - upgrade to a different house, especially in today's low interest rate environment.
If you can't imagine 6-8 weeks of renovations but really like where you live, perhaps consider doing a real nice job of finishing your basement instead and make the upstairs into just 1 or 2 continuous giant rooms - they would be very wide with only 1/2-3/4 width being head height, but still giant, without having to do any structural work.
Other alternatives include adding a new addition to the house, or adding a couple of large (say 3 joist spans wide - a bit under 6') dormers on each side in the existing roof. An addition you are of course talking several months work duration, but lesser disturbance because largely outside your current building envelope. Wide dormers are still not cheap, but doable and should not take over a week or so after the windows come in.
Another alternative would be a limited-foundation Florida room, depending on what the weather is like in your area in the winter, making that your rec room or living room and convert your existing rec or living room to bedroom space (being careful to do it in a "temporary" fashion, so can still be used as rec room or living room by a buyer when you sell your house).
One other thing that is common in Cape Cods and finished attics is to actually block off the low headroom areas with stub walls along the side, possibly using some of that low headroom area for built-in cupboards and dressers and such - see web images and videos on Cape Cod remodels. If you do that, you need EXPERT help on the ventilation, so you do not get dead air back in there and associated mildew problems. If you do that you HAVE to leave space to get back in there for inspection or make the stub walls as removeable panels, for access to remediate mold in case you ever get a roof leak, so a fair amount of your total floor space is lost, even though it does not cost you any head-height floor space.
Good Luck - hopefully a few other carpenter or general contractor types like Todd Shell will jump in here with their thoughts.