LAST FRIENDS continues and returns to the stories of OLD FILTH and THE MAN IN THE WOODEN HAT (and THE PEOPLE ON PRIVILEGE HILL).   Gardam does recapitulate many of the essentials, so the novel can  certainly be enjoyed without familiarity with the previous installments,  but familiarity does make for a considerably richer reading experience  (and since they're both wonderful works you should certainly (re)turn to  the two novels anyway).  
The story here begins with the service for Edward Feathers, 'Old  Filth' himself; with him, his longtime nemesis -- and, more recently,  neighbor and chess partner -- Terence Veneering, and the woman they both  loved, Feathers' wife, Betty, all dead there are just a few 'last  friends' of that generation and those experiences left.  Old Filth and Veneering were 'Titans' in their field -- English and  International law in the engineering and construction business.  Left now are only Dulcie -- a Dorset neighbor of retired Old Filth and  Veneering --, and another lawyer, Fred Fiscal-Smith -- who, despite the  fact that he had been Old Filth's best man (as he never fails to remind  folks, though he doesn't go into much detail about the circumstances  ...), "was never exactly one of us", as Dulcie notes.  Minor characters in the previous works, these two now come to the fore. 
Fiscal-Smith turns out to have rather more in common with  Veneering than first supposed, and it is Veneering's background that is  explored in LAST FRIENDS, along with the fate of the remaining  'last friends'.  The contemporary action follows Fiscal-Smith, who invites himself after  the funeral to Dulcie's for a stay (that does not go particularly well,  and sees him leave rather much earlier than he had hoped), as well as  Dulcie -- who, feeling guilty, tries to seek Fiscal-Smith out, which  also proves more complicated and adventurous than she originally  imagined.  The retrospective parts describe Veneering's extremely unlikely rise to  success, beginning with his unusual childhood. 
Few here are what they seem -- right down to matters of identity:  both 'Veneering' and at least the first half of 'Fiscal-Smith' are  adopted names -- and one of Gardam's running jokes is the misconceptions  folks have about Veneering's biography and many incidentals from the  past; Fiscal-Smith is one of the few who have much of a sense of the  facts.  
Raised in very humble circumstances -- his adoring mother sold  coal door to door --, Veneering was very lucky in many respects.  A very bright lad, he met the right people at the right time -- and made  the right choices, even as they often came just at the spur of the  moment.  He left his hometown during World War II pretty much at exactly the  last, best moment, and then made the right call when he was packed off,  with a boatload of other, younger children, to Canada when he was  fourteen.  He even stumbled across what were to become his Chambers at exactly the  right time, and on a rare occasion when he thinks he knows best and is  set to decline an offer, he is cajoled into hearing someone out -- and  finds himself set for life.  
The somewhat doddering old folks -- Dulcie and penny-pinching  Fiscal-Smith -- do adapt to the modern world, but also remain rather  apart from it, recognizing that they no longer quite belong or fit in --  Fiscal-Smith, who never quite fit in anyway, doubly so.  But they're also agreeably determined, and Gardam uses them very nicely  in her tale -- helped by some equally determined younger folk. 
The pleasures of the stories unfolded here include the many  quirky characters: it's not a no-nonsense English society -- there's a  lot of very amusing nonsense, too -- but it is one in which everyone  does know their place, and what they can do in that position (and then  does it).  The world -- and some of the characters -- may be rather rickety, but  they're determined enough to see everything through, and somehow things  work out, despite a variety of smaller, larger, and ridiculous crises.  
Enjoyable, too, are the connections Gardam weaves -- even the  twins Veneering encountered on the ship, young girls back then,  resurface nearby in the present -- and the layers of identity to the  various characters, especially from Veneering's childhood (and including  his father and the visitng nurse) that are uncovered over time are  particularly well-handled.  
A great deal here is presented casually and almost off-hand -- to  very good comedic effect -- and even as the story seems to be pulled in  different directions (and shifts between past and present) Gardam  manages to present a charming, cohesive sort of whole.  It's all a bit ramshackle (the characters, the estates, the stories),  but also good fun. 
One of the doddering old characters suggests that, long ago, the  wrong choice was made: "Perhaps fiction was a mistake, it has rather  fizzled out", but Gardam shows there's a lot of spark (and life, and  depth) left to it.