Let me start with English Birth Certificates. The one you have for your mother is a so-called "short" Birth Certificate which is given free to everyone registering a child's birth. The full version has to be paid for. It includes date and place of birth, child's name, mother's name and maiden name if married, father's name,occupation and address (if the parents are or married or if not married if the father is present to register). Most parents buy such a certificate. Those who don't either are usually very poor and can't afford it or wish to conceal the fact that the child was illegitimate and that there is no father's name. The birth of Charles Frederick Wilkinson was registered in Salford in the third quarter of 1903 matching your birth date of September3. Ordering this certificate from the GRO is probably your best course of action. The Reference is 8D 64
As to the church, don't be misled by the title "Church Administrator". This is not likely to be someone in a large shiny office with lots of equipment, more likely a lady in a cubbyhole with a photocopier who doesn't work full time and is paid little or nothing. I am sure the intention would be to get back to you but day to day matters such as getting the boiler mended are likely to take up most of her time. In addition the church may not even still have the Register. In so many years it is very possible that the Register has been filled and deposited in the Greater Manchester County Record Office
As to marriage announcements this would be more of a task than appears. Not everyone put such notices in any paper and a search would have to cover quite a number of titles, not just one. Possibly someone who lives in the Greater Manchester area (I no longer do) would be willing to go to Central Reference Library but it would be quite a job.
In short your best bet is that birth certificate because you should then be able to trace the parents back through earlier census records