Hypothesis: The geneticists increase the cytoskeletal density of the cells in your plants, creating signaling pathways like brains use. All the plants you mention contain sugars that can produce energy by chemical means, a way of storing energy taken in by photosynthesis and other reactions. Kelp genes and colonization by microbes form memory and assist decision making.
There already is a war. Chemical warfare by use of toxins and allergies in plants themselves or spread through pollen and spores. Biological warfare, especially by spreading parasites. Read about Toxoplasma gondii and learn to fear certain plants and cats and how they achieve mind control. Much of our pharmaceutical industry is engaged in fighting this war.
Communication is a tricky area. Sure, chemical signaling and all that, but that's usually interspecies and you need your mutants to make alliances and have a specific enemy. Thrumming the ground by various already understood plant mechanisms is possible but range and information density are low. Maybe take a hint from fibre optics and fluorescence and use light? Might also provide a means for humans to ally with the squashes, having someone discover a pattern in their light flashes and decode it. The plants don't have to develop technology, they just need to implement strategies of disrupting and destroying humans. Advanced kinases (see below) traveling spore-like could carry directives of successful methods.
The thinking part is not so difficult. Our understanding of "thinking" is being continually updated and neurons aren't the only game in town. Glia, microtubules, cytoskeletons (present in all cells) are all part of how information is transferred.
Think of computer chips: CPUs were power hungry beasts to begin with, like our brains; over time with changes in architecture the power requirement has lowered significantly. Glia in the brain helps in a similar way and microtubules and the structure of cytoskeletons represent nearly unpowered filament connections that act like neural pathways. Also think of present work on using carbon nanotubes for conducting signals; can be as efficient as copper and more useful in low power information systems.
Memory is required for thinking in terms of selective decision making and that ought to be simple enough to work out as it's basically a chemical addressing system similar to our emotional tagging.
Remember, we are basically half human, half microbe (plant-ish) by DNA count. So hosting is part of every complex creature and others have pointed out colonization by other bacteria, etc, would play a key role.
Protists, kinases, Paramecium. Protists are a category that both "animals" (single or multi cellular) and "plants" like algae and slime mold can fit into (quotation marks because by definition a Protist is not plant, animal or fungus). Kinases are proteins that are the directors, like symphony conductors, in organisms and use phosphates (like in fertilizer) to do this. Along with calcium fixing the pathways they can direct traffic. I put in a description below that fits into your story. Paramecium are great in that they have no neurons but seem to make decisions and even exhibit memory-like behavior.
I'm having trouble finding a link but Washington state grows a lot of carrots and they have been noted for anti-radiation properties. Around Hanford, where a lot of WWII nuclear work was done, there is much contamination and carrots were planted in an attempt to suck the radiation out of the ground. But here's a link to studies of plants around Chernobyl and Fukushima and a great line:
" activation of genes involved in DNA repair and of defence/stress responses following exposure of seedlings to radiation ".
-- (this is a bit of the technobabble from article cited above that might fit your story)
The CDPK family constitutes a group of kinases that are only found in
plants and protists. In plants, CDPKs mediate Ca2+ signals that
regulate a diverse number of pathways including cell cycle progression
and stress responses. The canonical CDPK is composed of an
amino-terminal serine/threonine kinase domain, followed by a junction
domain (also known as the autoinhibitory domain) that connects to the
carboxy-terminal calmodulin-like domain (Klimecka and Muszynska,
2007). The calmodulin-like domain typically consists of four EF hand
domains for Ca2+ binding. The autoinhibitory domain apparently
regulates CDPK by interacting with the kinase domain and acting as a
pseudosubstrate. Binding of Ca2+ to the EF hand domains relieves the
autoinhibition (Harmon et al., 2000).