What do you think I should do in this situation?

[Rufous Treecreeper]

I really think your Nan has given you her answer already with her negative feelings about DNA. Your relationship is much more important  than revelations that are likely to cause emotional harm to your Nan.
In this case I think you need to choose family over names.
Such a tough situation though, my heart goes out to you.

[River Tyne Lass]

I think your feelings matter as well, CurleyLocks.  I feel sorry for people in these types of situations when they are made to feel so guilty and shamed for wanting to know more about their past. None of us know what is around the corner.  Sometimes it might be too late to wait around until someone has died.  Don't be guilt tripped by anyone. 
If I were in your shoes, I think I would pursue things albeit discreetly as Pfig and Cell did.  In my opinion, just because there is an over riding need to find out more this does not mean anyone loves their immediate family any less. 
Reading your stories Pfig and Cell, I think you both handled things very well.  You had a win-win situation in the sense that you met your own needs in finding things out but you were also sensitive by not sharing information which might be unwelcome.

[Biggles50]

In "FAMILY TREE" which is more important to you "FAMILY" or "TREE"?

It is not that easy.

Family can be Biological or Genealogical and they are not necessarily the same.

In this case “Nan” knows of the DNA test and of the parentage question that it raises.

Her negative response is not going to “put the Genie back in the bottle”, she knows what is happening and her reaction is unfair on her family.

Educate her.

But

Certainly do not miss out on contacting relatives who share your bloodline, should they pass before Nan the chance of a relationship goes with them.

[Zaphod99]

I'm not suggesting breaking a confidence, but generally if you've known something for a very long time and that secret is eventually discovered, you have a lot of explaining to do. I think it is better to be open and candid now.

We should never be ashamed of things that our parents did.

Mrs Zaph.

[louisa maud]

Why should he or she have a  lot of explaining to  do, they   could say they  were protecting their  beloved Grandmother,  end of story in my opinion, they have to do what they feel is right for everyone , they have to live with it if they spill the beans to soon,  but as has already been suggested, grandma might know already and doesn't want to upset others,  works both ways.
Good luck with whatever choice you make

LM

[Cell]

In "FAMILY TREE" which is more important to you "FAMILY" or "TREE"?

It is not that easy.

Family can be Biological or Genealogical and they are not necessarily the same.

In this case “Nan” knows of the DNA test and of the parentage question that it raises.

Her negative response is not going to “put the Genie back in the bottle”, she knows what is happening and her reaction is unfair on her family.

Educate her.

But

Certainly do not miss out on contacting relatives who share your bloodline, should they pass before Nan the chance of a relationship goes with them.

Biggles have you ever had experience of this?  putting  it politely,   it is the OPs choice. Whatever they decide to do!
 I'll  never forget where you came in aggressively  to  a thread nwhen I found  my living parent  had had had  living half sibling  with  your  " the have right  to know."
It is an extremely  hard thing  to go through!!

what  is " educate her " ( as in their  nan) sorry this has totally  put  my back up!
!! Oh wow , how  derogative!!! You should  educate yourself  on  living  people ,close  kin which it affects , and not your own long  dead rellies - it's called FEELINGS,  LOVE and HURT

There are living people 
I am sure the OP knows  their family, and  thier  Nan better than you do. 
 Signing  off as I  am damn angry  with the bull that biggles is spouting off.



 

[Galium]

It looks to me as though you and your grandmother have different ideas about what a family tree is, and what it's for.

If your grandmother believes it is about social relationships, rather than biological - and that would appear to be the case, or she would want a DNA test - then perhaps you will have to accept that the biological truth isn't what she wants, or feels to be important to her research. 

It's really exciting to solve a genealogical mystery  for yourself (well done!), and of course you want to tell people about it. However, you don't seem to believe that your Nan would welcome the information you have.  Is it really necessary for this secret to be shared with her, or indeed with her biological father's family?  Who would benefit?

[Zaphod99]

I sometimes think the object here is to see who can start a fight quickest, a sort of genealogy version of Mornington Crescent, if you remember that far back!

Mrs Zaph

[Galium]

I don't have to remember far back. Latest edition of ISIHAC was last Monday