They are designed to have a reference to 0V ground. Many appliences with electronic components, particularly computers, can have quite serious problems when used on an old 127+127V supply in Europe so I doubt they'd be very usable on a US split phase supply either.
I'd say that there are actually very few British/European appliances for which this would be any problem.
As Schuko and most other European plugs are reversible, nothing can be designed to rely on the "neutral" being at 0V, and the switched-mode power supplies in modern computers will most certainly run just fine with either pole grounded or with each pole at 120V to ground.
