Louise,
Trade Directories (on line) have Nicholas Esposti at 176 Penarth Rd 1900/1901 - "Ice Cream Maker" + the 1901 Census. But by 1911 he is no longer in the business. Checked the Trade Directories in Glam Records Office, Cardiff today.
Two available for 1905 and 1906 - no mention of an Esposti or Nardini
Kelly's Directory 1910
Mardine (with an M), Lewis, Shopkeeper, 69 Carlisle St, South Splottlands
Western Mail Cardiff Directories 1915, 1920, 1922, 1924
Lewis Nardini, Confectioner, 69 Carlisle St, Moors
Western Mail Cardiff Directory 1929
Lewis Esposti, St Louis, Dalmuir Road, Moors
Nicholas Esposti, Confectioner, 69 Carlisle Street, Moors
Western Mail Cardiff Directory 1932
Lewis Esposti, 52 Mynachdy Road, Mynachdy
Nicholas Esposti, Confectioner, 69 Carlisle St
The next Trade Directory was dated 1949 - no trace of an Esposti/Nardini in the business.
So, Nicholas, having dropped out of the business after leaving 176 Penarth Road (between 1901 and 1905) returned to the business by 1929 when he's at 69 Carlisle St (previously Lewis Nardini's business). Explains why family members have memories of ice cream at Nicholas' post WW1?
Couldn't check Electoral Rolls - those pre 1949 are held at Cardiff Central Library.
However, the original enquiry was to put a name to the cafe - still outstanding. By coincidence, walking back from the Records to the City Centre I found myself on Penarth Rd - but insufficient time to double back and view No 176.
Hope this helps and good luck with the search.
regards