I understand why you want an easy answer. But you’re asking about the sound of three guitarists, each with a totally distinctive playing style, over the course of three decades. On our site you can find hundreds of pieces of gear, some used by one member of the band on only a single studio recording, and replicated live with different gear. If anything gives Radiohead a “sound”, it’s their approach to songwriting and arrangement. Or to be flippant: it’s modal mixture, especially the shift between major and minor, plus some sliding delay effects.
I can’t give you advice unless you listen to the band and decide what you like. To start, decide if you prefer Ed’s atmospheric style, Thom’s rhythmic style, or Jonny’s melodic style. Then look through the artist pages (in the sidebar of our site), and search for the songs with your favorite sounds. You can also find detailed pedalboards for every era, if you prefer to search by album. You should also check the “advice tab” in our sidebar — our post on modifying guitars for Lace Sensors directly answers your pickups question.
As for “necessary”, that depends on you and your goals. You can play Radiohead’s guitar parts on any traditional electric guitar (though like Radiohead, you might need a pitch-shifter for a few extra high notes). They don’t use 8-string guitars with extra frets or anything like that.
Do you want to look like the band? Do you want to get the sound of a specific era, or a specific song? Do you want an instrument that’s versatile enough to cover a wide range of sounds for live shows? You can buy an early-1990s Telecaster Plus V1, but you have to learn how to fit your sound with the rest of a band. Jonny did that just fine when he swapped his Tele Plus for his Les Paul Standard at Radiohead’s Haiti Benefit show. Radiohead are big enough to have have the luxury of touring with a bunch of amps and guitars, but they’re quite capable of getting similar sounds with less gear.
And to be clear, nobody in Radiohead has used the Lace Sensor “Blue, Silver, Red” Strat set. Jonny’s 90s Telecaster Plus V1 has its original Lace Blue and Lace Dually Red pickups, while Ed often plays 90s Eric Clapton Signature Strats with Lace Gold pickups. The Blue and Red pickups are humbucker voiced, while the Gold are single coil voiced. That should give you an idea of their respective preferences.
A photo of Jonny during Radiohead’s Haiti Benefit gig on January 24, 2010. Performed while the band was recording in LA for The King of Limbs, Jonny didn’t have his Tele Plus or his Eighty-Five. So he instead used his Les Paul and a Vox AC30 for many older songs.