When Radiohead play live, all synthesizers are DI. The output of the synths is processed by the FOH mixing desk, then played through the PA speakers. The same is true for Thom’s solo shows and the current The Smile tour. However, keyboard instruments are mic’d, namely the Kemble upright piano and the Rhodes electric piano. It’s also worth noting that on more recent tours, Radiohead’s FOH engineer Jim Warren has used plugins to process the band, so he may run the DI output of the synths through speaker or amp simulators on certain songs.

A screenshot of Jonny playing the Sequential Prophet-5 synthesizer during Speech Bubbles at Amsterdam’s Paradiso on May 27, 2022 (karindgr). The pedals sitting next to the keyboard are a Peterson StroboStomp HD tuner and an Earthquaker Devices Avalanche Run Delay/Reverb.
On the recent The Smile tour, the amplifier beneath the Sequential Prophet 5 Rev4 is in fact part of the Rhodes Suitcase electric piano. Radiohead have brought a Rhodes for every tour since 1997, and they always bring its amplifier cabinet, so clearly they consider it to be a core part of the Rhodes sound.
So far on the Smile tour, only Jonny has played the Prophet 5, so may be his synth. They probably brought the Prophet so that Jonny could stay in his part of the stage during Speech Bubbles. That’s the only song where Jonny plays a synth with his hands — though he plays his Oberheim SEM synth using a set of MIDI foot pedals on a bunch of songs, and on this tour he even started using the pedals during The Opposite. The Oberheim is also connected with a DI box to the FOH mixer.
During The Smile’s shows at London’s Magazine in January, Jonny played Speech Bubbles on the Dave Smith Prophet ’08 that is now part of Tom Skinner’s area of the stage. In a January 28 rehearsal video for the Magazine shows, you can see a set of six DI boxes under the music stand for the band’s synths.

The screenshot is blurry, but you can still see six Radial J48 Active DI boxes on the bottom of the synthesizer stand. Four of the boxes are for the four voices of the eurorack synthesizer case. The last two boxes are for the stereo outputs of the DSI Prophet ’08 synth (after it runs through a Strymon Timeline delay).
Obviously, using an active DI box per channel does get expensive. For those with a lower budget, passive DI boxes can get the job done too. Options like the Radial ProD2 offer multiple DI channels in a single box, and the channels can be used for anything, not just for stereo inputs. A passive DI can cause tone loss when used directly after a passive instrument like an electric guitar, but that can be solved by putting a buffered pedal between the guitar and the DI box.
Studio Recording
In studio, they use a combination of DI and various amplifiers and speakers, all just depending on the song. Photos suggest that they tend to prefer DI sounds, but Nigel may re-amp certain keyboard parts during mixing (a fancy way of saying he plays the DI recording through a mic’d speaker).

As an example, here’s a photo of Jonny in the Round Room at Tottenham House, from during the recording of Radiohead’s In Rainbows. Jonny has the Baldwin DS-50 amplifier running into a Ashdown ABM 300 C110 combo bass amplifier. However, Nigel is not mic’ing the amp directly — instead, he has a Neumann microphone pointed at the ceiling to record the natural reverberation in the domed marble room. He’s likely also recording the direct output of the keyboard, but it’s hard to tell with all the wires. Jonny also has a pair of Shure MDR headphones connected the Mackie 1202-VLZ3 mixer, as we discussed in this post.