Dials, for the most part cannot be cleaned. I am not talking about ceramic (vitreous enamel) dials, found on higher end wristwatches and pocket watches, but the vintage paint or coloured dials with applied markers type found on many brands.
That patina is caused by many things. Radium burns, smoke, water vapour, corrosion of the underlying metal, oxidization of the paint or dial finish, heat, UV light, adhesive decay, off-gassing of various components of the watch, e.g. plastic rings, certain old lubricants, whether or not the movt was ever 'dipped' (We NEVER dip) and the list goes on.
On a silvered dial you are seeing the oxidization of the silver compound, possibly also reaction to sulphides in the environment, and to attempt to clean these butler finished dials invariably ends up in disaster. That is not to say you cannot attempt to clean a dial, but you must be prepared to ruin it. For this reason, I do not clean the dials beyond some light surface debris removal. There is an old watchmaker's trick of using a soft boar bristle brush and lemon pledge, but this is also hit and miss, and can and will ruin a dial also.
The discolourization and 'dirt' is actually oxidization, and the breakdown of paints on the dial, adhesives on the markers, radiation, off-gassing of petroleum based or organic watch oils used in the past, reaction of the metal or paint to environmental pollutants and incursion of water or humidity.
We do offer complete dial refinishing service, and this is the only part of the restoration we do not do ourselves, we have specialists who do this. They have many of the original pad printing dies, and do the dials the way there were intended, fonts, logos, appliqués, guichet frames, you name it.
This process is not cheap, especially if the dial is for a valuable collector watch, which must be reproduced exactly. For my own watches, the dial has to pretty horrible before I will refinish them, but it really depends on the watch, and often how the dial was made and how the value may be affected by any refinishing. We usually have several examples in our display case should you be interested in what the finished product looks like, and will always be able to provide an accurate estimate for the job.
I should also mention it is unwise to attempt to clean your clock dial for many of the same reasons, but silvered dials cannot be polished, and to attempt to do so with Silvo or some like product, is a sure way to strip the dial completely. Dial silvering is a chemical process which we sometimes undertake to restore certain vintage clock dials. In the old, lamp or candle lit rooms, silver was the most visible dial in that type of light.