Hello - I've been trying to remember the words of this song too. A local pianist in my area used to sing it & sung it at our wedding after my mum had left, of course. I will contact him & come back with his lyrics but here's what I remember, in the meantime.

I love my mother in law
She's nothing but a bloody great whore
She haunts me day & night night night
I cant do nothing right right right right

Oh how I love,
how I love her
How I love my mother in law.

She's fine she's dandy
She makes our dog feel randy.
She's coming round to to stay,
Isn't it a pity she's only one titty,
& she's in the familly wa-a-hay (way)

Oh how I love her,
how I love her,
how I love my mother in law

Tonight I'll grease the stairs, stairs, stairs
put tin tacks on the chairs, chairs, chairs
I hope she breaks her back back back
cause I like wearing black black black.

Oh how I love her,
how I love her,
how I love my mother in Law.

One night in Gay Paree
I paid five francs to see
A great fat cheree
Tatooed from head to knee
And on her jaw was a picture of a prisoner of war
And on her back back back
Was a picture of the union jack, jack, jack jack,jack
and up & down her spine was the ???? in line
and on her big fat bum bum bum was a picture of the rising sun sun sun sun
And on her fanny was Al Johnson singing Mami

Oh how I love her
how I love her
how I love my mother in law.

Unsure if the version I knew included the Sinatra lyric as I'm sure I have missed bit's out. The pianist I know this song from was a big Max Miller fan I don't know if this was where it came from but possibly.