As a previous poster said, somewhere like the Ancestry website is a good place to start.
Do I understand this correctly. Your great granddad had two children with a woman that he wasn't married to? (At first, I thought he was married to your great grandma but then your great grandma had an affair with someone else?)
"...trace my great grandma"
Do you have an idea of their rough dates of birth and which part of the country they might have been born in? Finding their birth certificates is the thing that you really need to do as that will give the mother's name.
The problem is that, in this situation, the space for the father’s name will invariably be left blank on the birth certificate. The child was usually registered under the mother’s maiden name unless accompanied by the father, in which case both names would appear on the certificate.
You my get lucky and find that your grandma's birth certificate shows your great grandpa as the father. In which case, all done. But what if that doesn't happen?
So, what you would need to do then is to search the records for the area that they were born in. It really does help that there are siblings, it makes it easier.
For example, let's say that your grandma was born in 1920 and her sister was born in 1922. They were both born in Newcastle (this is just an example). Your grandma's name is Ivy and her sister's name is Florence.
So, what you would need to do is to find all the Florences born in Newcastle in 1920 and all of the Ivys born in Newcastle in 1922.
You are then looking for any matches where a Florence and an Ivy born in the correct years both have the same mother's maiden name.
If the first names are quite common then you may get more than one set to look at.
Once you have a list of Florence and Ivys with matching mother's maiden names you then need to look at the birth records to see if any of them were unmarried mothers (this will usually be shown by the mother's maiden name being the same as the surname of the child or no maiden name being given).
If you just have one pair of Florence and Ivys that are like this then it is likely that these are the correct people. If there is more than one then you will need to do a bit more research to identify who are the correct sisters.
I would also recommend that you think about getting a DNA test done (they're normally about £80 but are often on discount at £60). You will likely get a number of people related to you who are not descended from your great grandpa or your other great grandparents.
These people will be related to your missing great grandma. You can also use this to help identify who your great grandma was if the birth certificate route was causing problems.
"...and does it even sound likely?"
I don't know about likely, but it is certainly possible.
I certainly have something similar in my family tree.