Based on current deals at OCUK, this is one of the best builds I could come up with on your budget. Will definitely play any game at 1080p/1440p at very high framerates, any higher resolution, i'd probably recommend a GTX 1080. Should also handle streaming very well while gaming due to 6 core, 12 thread CPU. Can't really see any bottlenecks with these system specs based on your use case, however if you wanted to render videos for export onto youtube then i'd recommend using GPU acceleration within software such as Premier Pro to help with rendering times, you could purchase an 8 core CPU but it really wouldn't be worth it based on current pricing. Total of this build is £1285 based on current deals, but could go as high as £1350. This however does not factor in the cost of a pc case, I tend to exclude these out of builds because it comes down to user preference on the aesthetics of a build with things such as cases and lighting. Also periferals and a monitor are excluded from this build, A. because of personal preference and B. you didn't specify if you already owned these items. Finally storage is also down to user preference, i'd definately recomend getting an SSD as a boot drive to speed up the general use of the system and then something like a 1TB HDD for your games and some software.
i7 6800k - £430
Cooler Master Hyper 212 evo CPU cooler - £30
asus x99-e £210
corsair vengeance 16Gb 2666MHz (4x4Gb) - £110
msi Geforce GTX 1070 8GB - £380
corsair rm750x psu - £125
Final note, like the previous post said. Right now, it's probably best to wait for Ryzen from AMD (probably the most overused term in the PC building topic right now). Don't really know what they're going to come out with but it should shake up the market a little more and make them more of a competitor for intel.