You certainly can, Izzy. Many people do make enquiries like these as they simply can't get in to do the searches themselves.
The key thing is you know they exist as you have the info you have already! Get your info altogether: addresses, years, names etc. You can them help them in their searches this way to find all the possible info.
If there are more complete records for the family (and there should be given the arrangements made for the children) you hopefully will be able to find more details from these records. Also additional updates may have been added to the children's files and Jeanie & William.
Do you know when William Lyons died? Jeanie was later wasn't she? 1957 from the records. Did she remarry a Mr Rossini? Added: missed you previous post on this

All clear for that then.
Now, with Jeanie's mum being called Maria, does make sense that there would be a daughter called Mary/Maria.
Monica