Need help finding compatible bottom bracket and crank set for track bike My bike is this:

https://www.santafixie.co.uk/derail-black-fixed-bike.html

The BB measurements on the link are CH55 BSA 68/107.

I know literally bare minimum about bottom brackets and crank sets. I wanted to get this Miche Primato advanced track crank set and I need some ideas of what bottom bracket I could get that would work with this and my bike. (my bottom bracket currently has play in it so I need a new one anyway). 

Or if anyone could check my current specs and recommend a compatible BB and crank set (preferably 48t) that is good. I know there are other factors like chain line width but as I've given the BB length I'm thinking a same length new one would be the same chain line width? 

Cheers.
Like most bicycle components cartridge bottom brackets are highly standardized. Almost all bikes with threaded bottom bracket shells are ISO standard: 1.37inch x 24 threads per inch, drive side is left hand threaded. The ISO standard is also known as 'English' or 'BSA'. Obviously this is what your frame has.

There are two other parameters you need to know. Frame bottom bracket shell width and axle length.

Standard widths are 68mm and 73mm. The shell width is very easy to measure. Some cartridges will fit both a 73mm shell and a 68mm one with two 2.5mm spacers. 

Different models of cranks require a different length axle to get the same chainline. This is because the square taper in the crank arms can be in a different position along the axle axis (closer to or further away from the center of the frame. In general you can't put a different crank on a given BB and definitely expect the chainline to come out right.

What you need to do is know or measure what you chainline is, make sure you are buying a crank that gives you the correct chainline, get the correct bottom bracket axle length for the crank, and bottom bracket shell width of your frame.

If you are buying Shimano they make this very easy as they publish extensive specifications and compatibility. With other manufacturers it can be more difficult. I found some data for the Miche crank here, and I note that in actual fact it calls for a BB axle length of 107mm for a chainline of 42.5mm (which I assume is the standard for track bikes).

Another example, the manual for the FSA Vero Pro specifies a 110.5mm axle for chainline of 44mm.

You can of course modify the chainline of a square taper crank by getting a BB with an axle greater or less than what is specified (remember that to get 1mm of chainline difference you need 2mm difference in axle length).