UK bus and rail operator Arriva bought by US private equity firm

Probably not at all...

It will be less German (although truth be told it isn't much at the moment), it will get a new paint job, probably a new name, new uniforms, and life will go on.

I wonder if they will offer cheap package trips to American Baseball and Football (no feet involved except rarely) by train and stadium ticket?

I doubt much will change at all. Much of their work, like in London, is contract at fixed prices (to TfL). If they try the profiteering and offloading of US debt, favoured by US private equity firms, they will be outbid by lower priced companies.

Likewise for the rail contracts.

Much of UK public transit is now run by foreign companies so this is nothing new.

Outside London Arriva have quietly been disposing of bus assets anyway.

Probably no impact at all initially. Eventually though, it may increase prices. They did not buy it out of the goodness of their hearts. They bought it because they hope to make big profits. Simply business. Consider the consolidation of the airline industry and the hotel / motel industry in the U.S. The supply and demand theory sometimes means control the supply and demand the price you want.

Football (no feet involved except rarely)

Nigel, I just now made that connection! Duh!!! No wonder it's called football! Sorry, I am not a huge sports fan, although 2 of my grandkids play football/soccer, but now the name makes sense. What doesn't make sense is why America cribbed the name and used it for a full-body tackling sport.

What doesn't make sense is why America cribbed the name and used it for a full-body tackling sport.

Maybe because American football evolved from both rugby and soccer? Both being games referred to as football and the use of terms like ruggers or sockers referring to the various rules used or associations.