I have a MY 15.5 RRS 3.0 SD that has done 37k mile under LR extended warranty until May. What would you recommend the dealer checks before the warranty runs out, please?

The answer too your question is: "everything". But I would be extremely cautious. Extended LR policies are notably fussy over paying out. And if you do take out WD then WD won't cover pre-existing conditions. When is the MOT due? Consider getting a vehicle condition report from a reputable indy like RR London (Battersea) or Rover Tune (Reading), there are others. And then you are prepared as you put it through the main dealer process with your dwindling warranty coverage. If the MOT is not due, then I wouldn't do anything and just pray you won't need to claim on the LR warranty. L405 P400e Autobiography (MY2020)... Silicon Silver / Espresso
RRS TDV6 HSE Lux (MY2011)... Nara Bronze / Arabica - now gone!