I've always wondered this. Since dilithium crystals weren't discovered yet before Cochran made his very first warp flight he must've used something else instead. My first thought is that he used nuclear power to do it but wouldn't you need an energy source far more powerful than nuclear power to travel at the speed of light?
There is no official answer. Given the fact that it was built from nuclear missile components and had a very short duration it could be just about anything. Lithium crystals, some sort of magnetic focus like on a particle accelerator or fusion reactor, something not yet invented yet. There's tons of stuff in high-energy physics or particle-accelerator physics that could be extrapolated from. In fact I'd would wager it was something very much like a particle accelerator and it wasn't until a few years of intensive research had passed that other technologies were tried and discarded. How the humanity shown in First Contact arrived at what was shown in Enterprise could fill a novel. ... My head-canon is the warp-drive evolved from the EM-drive experiments that were quietly abandoned. A group of researchers figured out where that excess energy was going and with a high enough energy input, it could curve space-time enough to reach light-speed or just beyond. That energy input was initially a nuclear fusion warhead detonated in a fuel-rich sphere with the plasma directed into the nacelles by a brute force magnetic impeller. This single burst of energy input was just enough to bring the test vehicle to light speed. A more ambitious project using an actual fusion reactor to provide thrust and enough power to energize the nacelles was designed, but put on hold because of the widespread destruction caused by World War Three. However, Zefram Cochrane and his team would not be dissuaded from their efforts... and humanity rose from the ashes of defeat on the wings of a bird they barely understood.
Fusion and antimatter. You don't need dilithium to go to warp. You don't even need antimatter. You just need ENERGY for the warp coils. As we saw "Fusion Power" had already been an available technology by 2030's... With general timeframe between 2060's (WWIII post) and 2151 (Pre-Federation Earth) You don't need "Antimatter" or dilithium for warp, it's just more efficient. In "ST: First Contact" Lilly is Poisoned by Theta radiation; a common reactant of use by Antimatter use. So Phoenix was powered marginally by antimatter as an energy catalyst...... "Warp core" is just a nickname for matter-antimatter reactors. That suggests "warp plasma" is just a nickname for highly energy rich plasma. So you can have warp speed if you have another, although probably less efficient, way to generate warp plasma. Phoenix only achieved around Warp 1 (Lightspeed)
The Phoenix only maintained Warp 1 for a limited time frame before dropping to sublight... so, the ship itself could have used Fusion primarily, and some basic antimatter reaction (without dilithium) to give him the 'kick' for Warp - or just plain more potent fusion to keep him at Warp 1 for a brief period of time. As for the Warp core... I think the term can apply to any kind of power core (I don't think its limited to M/AM reactor as such). Warp can be generated using different power sources... dilithium and M/AM are but one of such methods. Other (more advanced and efficient) energy sources were also presented in Trek (TNG and later), and as we saw, in ST:Picard, Riker was using a term 'warp core' for Romulan Quantum Singularity power cores. I think the term 'warp core' is usually used in SF because the only FTL method available to most ships is Warp. But I think a 'warp core' could technically be any other power source which has the same or better efficiency than M/AM and dilithium.