No Mike, no keyless on the Icon, Icon Tech or Design models.
You can compare models on Toyota’s website to see what spec they have, click the Build your Corolla tab, click on Change Model & there are tick boxes to compare trims, once in the compare window you can change any of the 3 choices to any other Toyota available.
On the other models (hatch and estate) its available on the TREK and Excel trims, these trims are not sold by Toyota UK for the saloon, they top out at Design trim.
Toyota probably think the saloons are all going to end up as taxis/minicabs and don’t need high specs.
Might not be a big deal, the rest of the spec on the Design is very high & you do get push button start.
In life, it's not who you know that's important, it's how your wife found out.