Originally written in response to the following question, this should serve to answer any synthesizer recommendation inquiries.
Hello man, amazing work here! I was wondering What synth I could buy to pin down some of their most characteristic synth sounds (Let Down, Everything In It’s Right Place, Identikit, Staircase, Myxomatosis, Airbag’s “Martenot” [Korg prophecy]). I guess it would be the Prophet 08 but is there some other DSI cheaper ooption. Thanks :D

Thom and Jonny with a Sequential Circuits Prophet 5, Roland Juno-60, and Moog MiniMoog Voyager PE druing the recording of A Moon Shaped Pool. They also used their Dave Smith Instruments Prophet 08 and a Sequential Circuits Prophet 10 during the same sessions.
The synth on Let Down is a ZX Spectrum Computer. Everything in its Right Place is a Sequential Circuits Prophet 5. Staircase was a DSI Prophet 08. The bass on Climbing Up The Walls is an original Novation Bass Station, the bass on All I Need is Nigel’s Prophet 5, and the bass on Lotus Flower is a vintage MiniMoog Model D. The bass sound on Myxomatosis is a resonant monosynth layered with Colin’s fuzzy (Lovetone Big Cheese) bass guitar. The pads on Myxomatosis are a web of polysynths and chorus’d string-synthesizers (best heard 2:07-2:25, and in the last few seconds). Identikit’s “Broken hearts” section features synths panned left and right, probably a mixture of Prophets. The synthy choir on Motion Picture Soundtrack is a combination of Mellotron and ondes Martenot. Idioteque’s “synth” is a sample of a piece (Mild und Leise by Paul Lansky) created a room-sized IBM super-computer. The “synth” near the end of 15 Step is Ed’s Autoharp run through his pedalboard. As you note, the Martenot-like synth on Airbag (and Climbing Up The Walls) is a Korg Prophecy.

Thom and Jonny with an ARP 2600, a Roland SH101, and a Clavia Nord Lead 3 during the recording of In Rainbows.
It should be clear that no single synth can come close to replicating all of Radiohead’s synth sounds, or even their most distinctive.
First, there’s the fact that Radiohead have used a wide range of instruments over the years. Many synths (monophonic ones) are used only for bass sounds, while others (polyphonic) are used only for chordal textures. Others (string synthesizers, vintage samplers) are in different classes altogether. While most are subtractive synthesizers and have similar architecture, most have very distinctive filters. For example, the Bass Station, Roland SH-101 (used during the In Rainbows sessions), and MiniMoog Model D are all similar mono synths, but their filters sound radically different. By far the most significant factor in a subtractive synthesizer’s tone is the tone of its filter: regardless of how many oscillators, waveforms, envelopes, and extra features a synth offers, it will always sound like its filter. (This is part of why eurorack is so appealing: you can build a single synth with multiple filters cloned from vintage synths).
But even a digital synth with multiple filter emulations will have trouble replicating many of Radiohead’s “synth” sounds, because they often consist of other instruments layered with synthesizers, such as on Myxomatosis. And that doesn’t even cover the synthy sounds which they’ve sampled or created with other means.

Ed and Phil at Nigel’s old “The Hospital” studio during the recording of In Rainbows. On the right is an ARP 2600, while the stand on the right holds (from top to bottom) an Analogue Systems The Spawn and RS-15 Cabinet, a Korg MS-10, a Sequential Circuits Prophet 5, a PPG Wave 2.2/2.3, and an E-MU Emulator II. The RS15 Cabinet contains an RS200 Sequencer, an RS150 Seq/Switch, an RS340 Gate Delay, and an RS370 Polyphonic Harmonic Generator with RS375 Expander.
That said, Radiohead have used synths by one designer far more than any others: Dave Smith. Between the Prophet 5 and Prophet 08 (plus the Tetras which Thom and Jonny used 2012-2013), his synths have been on more tracks than those from any other brand or designer. While the filter on the Prophet 5 differs somewhat from the Prophet 08 and Tetra, the designs are similar and all use Curtis chips. As such, any one of these synths can do a pretty decent job of replicating the others.
The Actual Recommendations:
You’ll never pin down most of their most characteristic sounds with a single synthesizer, but you will be able to pin down the largest number of their sounds using a Dave Smith Instruments or Sequential Circuits synthesizer. The DSI Mopho X4 is a Prophet ‘08 with half the voices, but added sub-oscillators. The DSI Tetra (discontinued, but readily available on the used market) is the keyboard-less equivalent.

Jonny playing the old Sequential Circuits Prophet 5 during the recording of Man of War in 1998.
The one catch is that Radiohead almost exclusively use DSI and Sequential Circuits synths for polyphonic sounds (All I Need being an exception). The members of Radiohead generally use other synths for bass. For the past six or so years, Thom and Jonny have mainly used a MiniMoog when working with Radiohead. With Atoms for Peace, Thom and Nigel almost exclusively used an old Korg MS20 (best heard on Ingenue). Radiohead also had a Korg MS10 in studio during the recording of In Rainbows. And of course, Jonny has his modular synthesizers for monophonic (as well as polyphonic and rhythmic) sounds.
A good monophonic partner for a DSI polysynth would be the Korg MS20 mini. The MS20 mini is a relatively inexpensive monosynth capably of both traditional synth bass and lead sounds as well as experimental effects. Its pair of filters (high and lowpass) have a lot of character and are capable of smooth bass and intense modulation, and its pathway opens you up to modular synthesis. In fact, if you’re not too concerned with chordal textures, the MS20 would probably be a better starter synthesizer.

Nigel’s synth setup for the 2013 Atoms for Peace tour reflects the keyboards played by Thom and him during the recording of AMOK: DSI Prophet ‘08 for pads, Korg MS20 for bass and leads. The Roland A500 PRO MIDI controller above the Prophet is primarily used to trigger sampled string synthesizers for songs from The Eraser.