Yesterday I upgraded my City Hall and Academy to level 21 and was able to get two tier 4 troops (cavalry and archers) in just a few hours, instead of waiting a month for each of them. Here is how I did it:
I saved up a ton of discounted speed ups from the courier, I still have quite a lot leftover even after doing this.
Got a 15% research rune from Temple Guardians. If you don't have access to The Lost Temple, you can do this with a shrine or alter and still get a good bonus, the difference shouldn't be huge. The important thing is to make sure you gather the rune BEFORE you start the tier 4 research. It brought it down from 21 days to 18 days for me.
After starting the research, wait for all the alliance help before you start using speed ups, this will deduct a few more days off. For me this brought it to around 14.5 days. Then I popped the two 7-Day Speed Ups I had from City Hall rewards and had tier 4 archers pretty much right away.
After that I made sure to get the 15% rune again, then started researching tier 4 cavalry, waited for alliance help, and used a lot of 3-Hour Speed Ups I had saved.
My math is at 7/10 (8%), and my alliance tech at 2/10 (1%). If your research boosts are higher or lower than mine, you should be able to do this almost the same exact way, just with a few more or less speed ups. I probably still have enough speed ups to get T4 infantry now using the same method, but I ran out of resources so going to do it in a day or two probably.
If you have any questions let me know, good luck getting your tier 4 upgrades if you don't have them already!