Riding 50 miles (80km) in a day is a fairly modest goal. Doing it three days in a row is only slightly more ambitious. As a fit (as you will be when you do it) person to ride 80km, you could expect to take 3 to 3.5 hours if you rode hard, and 4 hours if you took it easy.
The difference is how you will feel the next day. For the actual ride, plan on taking it easy. Ride an hour and stop for a snack, take a few selfies, refill the water bottles, and ride on. So your 4 hours of riding would be spread over 5 to 6 hours.
For a modest and conservative plan, you can expect to be able to extend your rides by 10% per week. That would take you from 10 miles to 20 miles in 7 weeks, and to 40 miles in 14 weeks. If you can ride 40 miles (65 km) in one ride then you will easily cope with the "take it easy" scenario above.
Many people will be able to exceed that 10% per week guideline; experiment a bit to see how you go. But for planning purposes, it's better to be conservative and start early. The best laid plan takes real life into consideration. There will be times when you feel you need a break, when the bike needs work, etc. The worst that will happen if you start early is that you'll be even fitter.
Plan to ride twice a week for the first two weeks, to give your body time to adjust. Your seat will almost certainly complain when you start the second ride, but it should be ok after the next (3rd) ride. After two weeks, step up to 3 rides per week. Plan for one longer ride and two shorter ones. Mix it up a bit from time to time - skip a ride sometimes, ride on successive days sometimes, add an extra ride some weeks. The variety and spontaneity can help with enjoyment and also give insight to what you are able to achieve.
It would be great to get to 50 miles on successive training days, but you'll probably find that quite a challenge. It's enough to be able to ride 40 miles in one ride at your best pace, because on the actual tour you will be more gentle on yourself. Especially on the first day, and even more so in the first hour of the first day!
Sounds like a nice tour. Drive the route (by car or Google Street view) to check it out, and maybe join Strava or MapMyRide to see routes that other people use.