You should look into Y-chromosome DNA profiling
if there is presumed definite male-male line.
Look on the genealogy groups for advice on this.
Otherwise for general multilocus DNA profiles,
each generatiuon dlutes kinship by 50% and using
non-direct blood-line profiles you end up with the situation
that no link can be conclusively shown to exist and
at the same time not conclusively rule out a link

What they aren't telling you about DNA profiles
and what Special Branch don't want you to know.
http://www.nutteing2.freeservers.com/dnapr.htm
or nutteingd in a search engine

e mail nutteing2@quickfindit.....com (just one dot)

Or rather on rereading your piece, mitochondrial
DNA testing for the female line.
But equally try the genealogy groups for more info.

While it may be possible to determine with a very high (but not 100%)*
level of confidence that man A is *not* the father of child B, it's
almost never possible to have similar confidence that a particular man
*is* the father of a particular child.  If all you have to work from is
more distant relatives, accuracy suffers considerably.

If the curious grandson can persuade his father, his father's mother,
and his father's mother's husband to be tested, he may be able to find
out how likely it is that his father's mother's husband is his
grandfather. He may get an answer amounting to "almost certainly no"
but the best he'll get in the other direction is a percent probability.
Note that UK and US labs may make different estimates of probability
because one factor is the ethnic diversity of the relevant populations.
If the probability isn't high enough to satisfy him, he'll just have
to take his grandmother's word on the matter.

From an "Ann Landers" point of view, I think it's unfortunate that all
this trouble is being stirred up, just to satisfy curiosity, when those
directly involved settled it all to their mutual satisfaction perhaps
60 years ago.  It's hard to tell from here how much of this is being
driven by the grandson, the grandmother, or your mother, but I feel
sorry for your dying uncle getting dragged into it.  It sounds like
your mother has made a friend, whether or not she's the mother of your
cousin.  Perhaps they can expand their correspondence to other topics,
and your mother can decide whether or not to accept the her new friend's
word that she may have a nephew and grandnephew in England.

* (Note that while paternity testing services may make it sound simple
and definite, the advent of paternity testing has demonstrated yet
again that biological systems are seldom simple and straightforward.
It seems that genetic mosaicism and chimerism are more common than once
thought:  paternity testing has sometimes shown that the biological
*mother* cannot be the genetic parent of her children.  And while it adds
to the emotional distress of the situation, people do edit their memories
unconsciously, and after 60 years it's possible that the grandmother
could have edited out other candidates for the paternity of her son.
Those were frightening, desperate times in WWII Britain, and many people
did things they would never have considered under more normal conditions.)