Hello and thank you for your question,
I'm afraid that I do not know whether or not there was a requirement to register change of names at the time such that either your great grandfather or your grandfather was required to register the change of name. But I cannot see that it would matter significantly where your mother is alive and in the UK and presumably prepared to give evidence that the person named on your grandfather's birth certificate is the same as the person named on her birth certificate. For the purposes of immigration rule 186, which is that which would be applied, it is usually necessary to obtain, as you have, all the necessary marraige and birth certificates to show the connection.
Were the ECO or as the case may be the UKBA to dispute the connection between yourself and your grandfather, no doubt your mother would be able to provide evidence on this issue. And it would be best for her to do so prior to the application being made.
Even if he, unlike perhaps your great grandfather, was required to register the name change on reaching majority, the important thing to establish is that regardless of him not having done so, there is sufficient evidence to establish that the person so named is more likely than not to have been the same person named on your mother's certificate.
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