We are proposing to visit Italy at the end of May. However the appointment with Italy is not possible till the end of May for the visa interview. We are getting an appointment from the French embassy for a visa. If we get the visa through the French embassy, can we go to Italy without touching France? We do not intend to visit France at all. From Italy we propose to go to the UK.
Unfortunately, this poses serious problems.
You are expected to apply for a Schengen visa from the country of your "primary destination." This is the location of the "main purpose" of your trip (if there is one, e.g. you will attend important business meetings in one country) or simply the Schengen country you will stay the longest. According to the rules, a consulate is not supposed to issue you a visa unless your application conforms to these rules.
When you apply for your visa, you'll need to supply supporting documents that provide your itinerary, generally including details of where you will be staying. If you are honest, the French officials will see these documents and determine that you are not staying in France, and your application will be returned, because you've applied to the wrong place.
Alternatively, if you lie and submit a false itinerary for France, you might get a visa, but you could be caught at the border having obtained a visa by deception. Lying on a visa application is really not recommended: it can lead to an abrupt end to your trip and/or great difficulty ever getting a visa again in the future.
What you could do instead is rearrange your trip such that you actually spend more time in France than Italy (or an equal amount of time and visit France first), such that the French Consulate is the proper place to file your visa application. Or reschedule your trip for later in the year after you are able to get a visa.