There is multiple setup for build directly in UE4 editor where you can lower the size of files. Current setup you use is probably target for debug. If you choose build for shipping, you will get much smaller files, engine itself is around 47MB, then there is small loader, your game logic (size varies ofc) and your game data (again size varies).
I wouldnt recomend any other third party minification software in case of UE4 engine, not sure if it would be compatible with asm.js magic they use. I belive right now it is as small as it can be, if you consider all the features UE4 engine provides and compare the size to native applications...
Also notice that files are being gziped/bziped for transfer, according to my browser Im transfering 30MB for my app, but total size on disk is ~120MB.
Another thing that might help, start the most empty project and not the project with starter content. It will lower the size for app data.
If you give me few minutes, I might post here some screenshot...
There is first place where you can set build for shipping:

There is another place:
