Well, Japan had a history of ordering ships from Britain in the pre-WWI era; and the Kongos were part of this, though as you say the later ones were built in Japan proper. The next designs, which were the Fuso class and the follow on Ise class, are clearly built using similar principles, but with a different purpose.
Nagato has differences, but follows a similar line of reasoning from Ise->Nagato that existed in Lion->Queen Elizabeth, though this was a trend that was paralelled in both the US and German navies too. Its probably the first Japanese BB that I would consider to be truly of their 'own' design, including Japanese designed weapons, armour layouts, etc.
The ships cancelled by the Washington Naval Treaty which were part of the 8-8 plan (Tosa class, Amagi class, Kii class, No13 class) are clearly different from the designs pursued by Britain at the time (G3, N3, other variants). If one studies the range of designs suggested by various nations in this era it becomes clear that a lot of similar concepts were looked at, but everyone ended up with slightly different choices. I have no doubt that there was some measure of awareness of what the other naval powers were doing, but in the end everyone has their own requirements and ideas of what is needed in a ship.
From the WW2 era, Taiho and the follow on G14 design are interesting ones, since they are very similar in design to some of the UK armoured carriers, which may have had some influence on their creation.
Like Warspite said, a lot of the naval tradition and procedures in the IJN had their roots in RN teachings.
< Message edited by JuanG -- 12/25/2011 7:09:33 PM >