I'm not sure where you're finding those tree matches.
Looking at "Search all member trees" under "Search" at top right of this page, the only Julia Riordan 1854+/-1 born in Ireland is your own entry..
The only other Julia Riordan born 1855 on any tree on GR is one born in Canada, so I don't see how you can be getting "a couple of matches".
Be aware that "Tree matches" from the purple star beside names on your tree are only people who happen to have the same name and year of birth as your relative. Place of birth is not taken into account by the GR algorithm, so that "matches" are often not related to you at all.
Also be aware that when you send a message to any "owner" of a tree on GR, there is no guarantee of a reply. GR never deletes a tree, so a tree owner may have died, or changed email address, or for many other reasons may not receive your message, or be unable to reply.
Always send a proper message explaining your interest - don't just ask to see their tree, as that's likely to be ignored.
You say these "matches" give Mary Riordan as Julia's mum - how do you know that ? You can't see anyone's GR tree without the owner's permission.
When I say matches I mean the little purple star you get on the corner of the entry. I get purple 7 I click view and get 5 matches for Mary Riordan to those 5 matches I sent the following message:-
When Julia Riordan (Reardon)B July 20 1855 D 1935 registered her daughter Julia Cahill B 1889
D June 27 1915 Mary Riordan was listed as her grandmother, I wonder if we have the same Mary Redmond. I look forward to hearing from you Take care and have fun Shell x
I don't just ask to see family tree that just rude.
I am aware that people can die and there trees remain I have matched with my Great Aunts family tree and she has sadly passed away years ago. It is frustrating as she would be a gold mine of information and it would have nice to share an interest with her.
I am finding it frustrating that these boards don't allow you upload information.
I am also a little annoyed GR don't ping you when you get a reply to a thread lol
I like Facebook you get a message to say someone has posted on thread or a thread you are following.
Take care and have fun Shell x
GR doesn't have the resources to notify you of replies - and it's not exactly a hardship to come back and look
It's amazing how many people do just ask to see a tree without any explanation!
I'm afraid your message to the "matches" for Mary Riordan is rather confusing.
"When Julia Riordan (Reardon)B July 20 1855 D 1935 registered her daughter Julia Cahill B 1889
D June 27 1915 Mary Riordan was listed as her grandmother, I wonder if we have the same Mary Redmond.. . ."
A woman should be listed on a family tree under her maiden name, so unless Mary was unmarried, or unless people have just listed her as Riordan because they don't know her maiden name (although they shouldn't do that) , the Mary Riordan "matches" are unlikely to be her, unfortunately.
Mary Redmond? Did you mean to say Julia Redmond or Mary Riordan?
Also, young Julia's birth was registered under the name Riordan, not Cahill.
However, hopefully an actual relative will know that she was known as Cahill later, and will understand what you mean.
It will be interesting to know if you receive any positive replies. Good luck!
As so many Irish records no longer exist, it may not be possible to find out any more about Mary.