This answer tries to establish that Dilithium was part of the Romulan propulsion drive, too. Even if it weren't, it seems that the Romulan society as we know it no longer exists at that time - but that's out of scope for this answer.
The singularity is the power source, but Dilithium is a controlling agent
Now this is a bit fuzzy, since in a few episodes, it is stated that ships "run on" dilithium - but it seems that ships need antimatter as fuel, and dilithium to control the reaction.
And in case of the romulans, it's the power source that has been changed, it stands to reason that dilithium plays the same or a similar role in their propulsion systems.
Dilithium is not the power source, but a controlling agent
This is a bit fuzzy since it has been stated a few times that ships "run on" dilithium. But more specifically, ships need dilithium to run, but not as the power source itself.
It seems that in case of a warp drive, you need a stable dilithium matrix and some antimatter, which is the actual fuel. This is confirmed a few times throughout TNG:
LAFORGE: This is my gift to the Victory's Captain Zimbata.
DATA: Most unusual.
LAFORGE: I served with him an ensign. Sure wish he'd been in command of this Victory. Wind and sail, that's the proper way to move a ship.
DATA: But, Geordi, your Starfleet specialty is antimatter power, dilithium regulators
LAFORGE: That's exactly why this fascinates me, Data.
(TNG: Elementary, Dear Data)
More specifically, Dilithium seems to help "aligning" the matter/antimatter flow:
WESLEY: Him Mom. You're back early.
(She looks at the computer monitor he's studying)
WESLEY: Physical sciences class. We're studying Doctor Channing's theory on dilithium crystals.
CRUSHER: Tell me about it.
WESLEY: Really? You never seemed that interested in warp theory before. Doctor Channing thinks it's possible to force dilithium into even more useful crystals. If as shown here, matter and antimatter could be aligned even more efficiently.
(TNG: Lonely Among Us)
It's stated outright that Dilithium is not fueling the drive in one episode of TNG:
RIKER [OC]: Now, what are the possibilities of warp drive?
LAFORGE: Not good. There are only a few dilithium fragments left in the holding clamps. Even if we had crystals that were intact, there's no anti-matter to fuel the drive.
RIKER [OC]: Any recommendations?
LAFORGE: No, sir.
WESLEY: We haven't got a prayer.
(TNG: Peak Performance)
It's specifically stated that the singularity is the power source of Romulan star ships
O'BRIEN 2: They've hit the fusion reactor. Main power's down. Defence systems are offline.
(Our O'Brien collapses in agony.)
KIRA: Try to re-route phasers through the emergency power couplings.
O'BRIEN 2: Miles, you've got your answer. The quantum singularity that's been orbiting the station. It's the Romulans. Are you listening? The Romulans use a confined singularity to power their warp core. That's what we've been detecting. That's what's causing the time shifts. You must go back and tell them to put up the shields and stand by for an attack.
O'BRIEN: I can't. I can't get through another shift. It's the radiation poisoning.
(He gives the armband to his future self)
O'BRIEN: You go. Just activate it. It's set for the return trip.
O'BRIEN 2: Me? I don't belong there.
O'BRIEN: I'm you, you're me, it doesn't matter. Just go. Go!
Similar things are mentioned in TNG: Face of the Enemy.
Romulans are intereted in dilithium, too
Even though it's never explained what they use it for, "Star Trek: Nemesis" tells us that there are Dilithium mines on Remus:
DATA: As you can see one side of Remus always faces the sun. Due to the extreme temperatures on that half of their world, the Remans live on the dark side of the planet. ...Almost nothing is known about the Reman homeworld, although intelligence scans have proven the existence of dilithium mining and heavy weapons construction. ...The Remans themselves are considered an undesirable caste in the hierarchy of the Empire.
(Star Trek: Nemesis)
Apart from these mines, there is also the fact that Romulans successfully stole the schematics for a dilithium chamber in TNG: The Drumhead. In both cases, it's never elaborated upon what they are actually using it for, but to me, the picture emerges that they need it as much as anybody else in the galaxy.
Lorca told us
This is what Captain Lorca told us:
LORCA: Gentlemen. Every starship in the galaxy, Klingon or Federation, runs on dilithium crystals. If we can't protect Corvan, the war is lost. So, can you fix the Lieutenant's inability to get our ship to go where it's supposed to go?
(DIS: The Butcher's Knife Cares Not for the Lamb's Cry)
There are many reasons not to believe this quote - it could be hyperbole, it could be referring to Klingon vs Federation only, Lorca is not trustworthy and so on - but I still included it because it's the only quote I found that deals specifically with dilithium and space travel in general.