Tyres (spelling important since we introduced them into America) are covered in depth within the EASA part 66 B1.1 licence exams (part 147). Topics include basic maintenance, construction, fabric weaves and layups & bead construction.
Batteries are covered in depth too - even down to specific electrolytes and how to service the things (once you get to the cells)
Wood and fabric was removed from the syllabus a few years ago since the B1.1 only entitles the holder to work on complex aircraft above 5700kg in weight. (Which is 'quite heavy' in Americanese). It has however, made a small comeback and is mentioned in the 'Materials' module as an introduction to Composite materials. As EASA eases in the light aircraft licence, Wood and fabric will play a bigger role.... unless of course you guys over the pond still operate fabric and wood airliners....
The European and Antipodean licences are specific, but cover a huge range of subjects including avionics and electrics. For the heavy duty black box changing, we still have need of fairies and dog f***ers... :P