I help do the washing up.
Would it be correct to say "I help WITH the washing up." too?
This one, 'I help doing the washing up', doesn't sound quite right to me, Lenka.
I'm not a BrE user, but are you sure, Bob, that this is used in BrE?
With the addition of 'in' or 'with', it sounds more natural to me.
'I help in/with doing the washing up'.