Hi Clare
I know for certain in my case up to at least 1885 it seems there's no ships manifests for our ancestors coming to UK from Italy. As Tomo said most of them crossed France then got a boat there to take them to the London area of UK. As it was a short passage at that time the ships' captains did not need to and did not usually keep a record of the passengers travelling to UK.
I have searched for a long time about this and even had a friend exam records at Kew and always got the same reply.
Only on one occasion i was told if I knew the names of the passengers, or their proffession and the port they arrived at I would perhaps stand a chance of finding this out. No results form this either Not sure if records had to be kept by the 1920 though so you may have a bit more luck there.
The only way I was able to get a rough idea of when my grandparents arrived in UK was to check the Family History Microfilms to discover if they had children born in Italy before they came here and find their birth records and then check the year they had family born in UK that let me find out they arrived here between 1885/1887
Now if they had been going to USA that would be a different story

Dont give up and if you find anything please let me know
Emmy