The following is all entirely your opinion so feel free to fire away.
For me, I'd probably use elements of The Crystal Egg (which some see as a 'prequel' to The War of the Worlds) and Star Begotten - which treats The War of the Worlds as a novel but whose events are sometimes used for other works that combine/continue the events of The War of the Worlds. As for what the story will be like, I can imagine it been similar to Stranger Things.
For my own idea story (which I'll call War of the Worlds: Renegades), I'd say that when the first War of the Worlds ended with the defeat of the Martians thanks to Earth's bacteria, the Martians would still set up a new base on Venus whilst starting "harvesting" missions on asteroids in the Solar System to get what they need to survive without fear of attack or disease.
In the meantime, they begin experiments with their own biology to try and get around the issue of illnesses, whilst also upgrading their current technology and working on new weapons, technology and methods of war to prepare for the potential day that Britain and/or other nations on Earth launch their counterattack.
In the years that follow, humanity does make significant advances with some of the Martian technology - including the secrets of supersonic flight, the creation of new alloys, the proliferation of Red Weed as a substitute fuel source and the successful construction of working Heat Rays. However, some of it ultimately remains a mystery to even the best scientists - including the Viridigen gas, the black smoke and the arrangement of the crystalline wafers in the Digging Machines that would otherwise aid in the future development of automated machines.
By the 21st century history has mostly played out like before, with both World Wars, Britain's Empire breaking apart and other nations becoming more powerful - discovering alternative technology/weaponry that partially or completely nullified Britain's earlier advantages granted by the Martian tech.
Meanwhile on Mars, the Martians have greatly upgraded their technology - with spacecraft capable of hiding from humanity's methods of detection making journeys to Earth to abduct humans for use in their experiments. With attempts to immunize their current bodies against the diseases of Earth proving futile, the Martians begin a new plan and use their vast intelligence to create a twin pair of human/martian hybrids (who I'll dub "Aries" and "Athena" for this bit) with the eventual goal been to re-invade Earth and turn themselves and the human population into Human/Martian hybrids.
However, while Aries and Athena have gained the higher intelligence and psychic power of the Martians and the ability for their bodies to fight off Earth's illnesses, they've also maintained their emotions and sense of humanity. Seeing through their psychic link with the other Martians, they witness the historical first attack on Earth and learn to their horror of the Martian's plans.
Using their newfound creativity and enhanced intelligence, the pair manage to take what they need (including the 'Master' Crystal Egg and details of the Martian's new weapons of war) before making a pitched but successful escape to Earth in one of the Martian's spacecraft - crash landing in the English countryside.
There, they're eventually found by the authorities after been taking in by an elderly couple - the twins working together with the British Government to help Britain and the world prepare for the invasion to come (a homage to the Austrians and Germans who defected to the Allies during WW2 when the Nazi Party took over).
So what about you guys? If you made your own sequel to the War of the Worlds, what would you do?