As to the childish term
willy (I've never spelt it "willie", although the plural is "willies") - from way back in the mists:
Mary had a little lamb
She thought it rather silly
She threw it up into the air
And caught it by its
Willy was a watch dog
Sitting on the grass
Along came a bumblebee
Who stung him up his
Ask no questions, tell no lies
Ever seen a Scotsman doing up his
Flies..... That's enough of that!

I agree with Billy Connolly (a hero even for many English people) that the biologically correct terms are appalling

. Little boys (of all ages) have
willies, men generally have
dicks. "Dick" is rude, but not
that rude. We're back to seaside postcard humour with that famous English pudding
Spotted Dick - note that it's referred to as "Spotted Richard" only by jokers, never by even the most politically correct.

Similarly (Bill), there's rarely much, if any, embarrassment attached to men being called Willy or Dick. In that context they're more slightly affectionate diminutives, as in Willy Shakespeare, Dick Turpin and Dick Whittington.
Having said which, some of my work colleagues - not me, of course - had an unworthy chuckle at the expense of someone called Dick Long, whose emails presented his name surname first..
Fisk