Virtually all the stuff you buy from the US will have JIC or NPT threads.
In the UK we traditionally use BSP [some BSPT] but have used a lot of METRIC for the past 10 years. Most rams and valves you buy in UK will have, at least the option of, BSP.
Confusion is easy [this isn't strictly 100% technicallycorrect but explains well]............ if you are talking UK and you say 3/8 BSP that is a fitting that will be used with a hose that has a 3/8 bore. If you are talking US and you say 9/16 JIC that is a fitting that had a 9/16 thread form NOT related to a specific hose bore.
Unless you are just building a 12volt slow system you want to use fittings and hose that will give you a 3/8 bore all the way so regardless of what thread form you are using that is what you need to specify. If you are not SURE you ask "I want to go from this to this [show him the parts] with fittings and hose that keeps a 3/8 bore all the way".................
BSP [paralell or taper] thread looks like this

This is a 3/8 BSP fitting. Notice the concave cone and the flat shoulder for a dowty seal.
BSP threads are used in hose fittings and in port fittings [as you would screw into a valve block]. With a port fitting you need to use a Dowty seal to seal the oil pressure.
JIC looks like this

This is a 9/16 JIC fitting. Notice convex cone and no shoulder.
JIC is mainly only used for hose fittings. Port fittings will be NPT, this is tapered and you use PTFE tape on the thread to seal the oil pressure.

On right is a standard US inline check valve [ 3/8 NPT], on left a 9/16 JIC fitting the size US will supply for use from dump to rams.
Take a standard US dump valve. The port size will be 3/8 NPTF. The fitting you will get in a kit to go there will be an elbow with 9/16 JIC on the other end for a hose.
If you now run a hose from the dump fitting to a UK sourced shut off [on an accumulator for example] you will us a 3/8 bore hose [same size as the dump hose you get in the kit BUT be aware that they sometimes use lower pressure hose for dump hoses to save some cents!].
At the UK valve block you will have BSP sizes. You now have a choice of using a BSP hose fitting on the end of your hose from the dump [so you change everything from the on to BSP] or using a tee to the acc valve that has BSP on the branch and keeps JIC for the hose run. Personally I like to use JIC where ever I can.
Rus. I suspect that you have used a 3/8m to 1/4m [1BP0406] fitting in your shut off block, then a 1/4 f-f-f tee [ZFB04] with two 1/4m-1/4m [1BP0404] fittings ?
Really you should go for 3/8mBSP-9/16f JIC [1BJP0609] in the block then a 9/16 JIC m-m-m tee [ZMJ09] and use 9/16 JIC swage fittings with 3/8R2AT hose.