Toyota said domestic production fell in July, outweighing record overseas production that was driven by a strong recovery in Europe, China and the rest of Asia.
Honda blamed delays in receiving parts and logistics due to COVID-19 and semiconductor shortages for the reduced output at two domestic plants.
Toyota's global production plan for June stands at about 850,000 vehicles, a cut of about 100,000 vehicles.
The U.K. is the latest European target market for BYD in its push to build a presence outside China.
CATL already plans to build a second battery plant in Hungary as it readies production at its first factory in the region, in the central German city of Erfurt.