Certainly it would be courteous to inform the author's heirs of your intention, and possibly thereby gain their blessing for your book, but legally you don't need their permission or that of the publishers.
Characters per se are not specifically protected in UK copyright law*. If a character or a major plot device from the first book formed a substantial part of that book, reusing it in your book might amount to copyright infringement, but it sounds from your description that, except for the resume you mention, you will not be doing this. On that basis I think you should be OK to go ahead with your project.
However to gain some insight into how the courts have viewed this kind of issue in the past, take a look at the case known as the Da Vinci Code case
in which Dan Brown was found not to have infringed the copyright of an earlier book which covered much of the same ground as the Da Vinci Code
. While this wasn't about sequels, the principles are much the same. The Da Vinci Code case went to the Court of Appeal where, apart from some gentle criticism of the trial judge's rather unclear judgment, the CA dismissed the appeal. For the sake of completeness I am including a link to CA's judgment
but because they agreed with the trial judge, you don't really need to read it, unless you want to see further analysis of how the courts examine the issue of a 'substantial part'.
There is a second matter which, if you were publishing your book in France, might have particular significance, and that is the moral right
of the author of the first book not to have his work treated in a derogatory manner. I am not suggesting that your book would do this, but under French law the author's droit d'integritÃ©
is taken so seriously that even writing a sequel which was judged not the match the literary brilliance of the first book has led to a second author being fined, even though the original book was out of copyright at the time the sequel was written.
* In the USA this aspect of copyright has expanded considerably as can be seen from a long running series of disputes
about characters and details taken from the Sherlock Holmes canon of Arthur Conan Doyle. And even the Batmobile from the Batman TV series has been declared a 'character' protected by copyright. It is highly doubtful that such decisions would be found by the UK courts.