I have medium long hair that is curly/wavy and is totally frizzed out if I don't wash it and put a bit of silicone and gel in it to calm it down. To be honest I found that if I just stop and think about what I am putting myself through each day I can make changes that make my life easier.
If you have extra greasy hair think about getting a body wave perm. It makes the hair less greasy and will keep it from looking flat and limp.
If you hae dry hair then go to Sally's or beaty co and buy some Aphogee Gloss Theraly and use it after each wash. It will make your hair shine and give it bounce. Use only small amounts until you find the right amount for your hair.
If you have straight as a string hair then towel dry it as best as you can and then let it go dry by itself while you are getting breakfast and other stuff.
If it is truly something you cannot manage without 45 minutes of drying and styling then it is time to contact a prfessional for an update on hour hairstyle. It may be that just trimming it into a little better style may make things easier and let it look nice so you can go a couple fo days without washing it every morning.
When my hair was longer I rolled it up wet on large rollers and used a hood type hair dryer. The curl goes deep that way and you really just need to brush it out each morning and it looks the same day after day. If it looks a bit limp then hot rollers help fluff it back out.
Since you know how to roll your hair I think the best option would be trying to roll it and dry it in the rollers. Like an old lady at the beauty shop.
But my hair sounds similar to yours, mine is not as thick, and I used to take an hour every day to do my hair too. I had a hair dryer with a half round brush on it and as I dried each indiviual piece of my hiar it went on a roller to cool down. It was very time consuming. I discovered the Gloss Therapy and mousse/gel and my hair looks amazing most of the time. I wash, condition, towel dry a bit, put some Gloss Therapy in it (concentr4ating ont he ends to make sure they get enough) and then add some hair gel (I use Pantene or some toehr name brand) and then I bend over and fluff it a bit and I am off to whateveh I am going to do. I get compliments on my hair all the time. At first it was hard making the changes because I was so ingrained in to my hair routine. But I am so glad I made the changes. it's almost like my hair has gotten crulier as time has passed. Getting it trimmed into a better style helped too. It lightened it so the curl it had could do it's job.