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.
Thank you very much for the reply, to the comment above my nan is educated on genealogy and actively watches shows where the topic of illegitimate parentage are covered, She had no issue with me and my mum doing the DNA test she just seems quite reserved against it herself. 
There is no clear answer as everyone will deal with it differently but again its the mixture of guilt of keeping secrets from her but I also dont want to stress her in a more delicate era of life