From Wikipedia, but it goes to show what I mean:
"As part of the plan, design work on the P class was halted in mid-1939 in favor of the O class. Displacement was limited to 30,000 t (30,000 long tons) in the new designs so that the length of construction would be shortened from the normal four or more years that a battleship required, to an estimated three
[5] to three and a half years.
[6] Required characteristics for the battlecruisers was a displacement of 30,000 t (29,526 long tons), a
main battery of six 380 mm (15 in) guns, a secondary battery of
dual purpose guns, a top speed of 34 knots (63 km/h; 39 mph), a range of 15,000 nmi (28,000 km) at 19 knots (35 km/h; 22 mph), and enough armor to counter the 203 mm (8 in) guns of heavy cruisers.
[7]"
This is likely to be our "30k ton lightweight" version of KW45 and 50, interest being revived in battlecruisers in 1938 and design work seriously beginning for them in 1939, same as described in the RM 20 text (January 1939).
I would like to add that, just because these designs come from a private shipyard, does not mean that they were not seriously considered. Spahkreuzer 38 - a class of 22 planned ships under the Z Plan, 1 laid down and several engine sets completed and more ordered, began its life as a Blohm and Voss alternative to the Type 1938A destroyer series and was taken in and subsequently modified.