All nuclear reactors need that "first neutron" to get things going. It doesn't matter how fast the first neutron is going either, since the moderator will slow it down ("thermalize" it) just like it does with fission neutrons.
The "first neutron" can be supplied by a variety of neutron sources -- isotopes that emit neutrons either spontaneously, or when bombarded by other forms of radiation (e.g. alpha, gamma) due to other isotopes nearby. For example, a relatively small quantity of californium-252 will spontaneously emit millions of neutrons per second.
Once the reactor is started, the external neutron source is removed and stored for the next time it is required.
Interestingly, in a reactor core that has only recently been shut down (ie., the fission chain reaction ended by insertion of control rods), there are often enough neutrons emitted from fission products in the irradiated uranium fuel, that the chain reaction will initiate just by raising the control rods. In practice the rate of neutron multiplication may be too low for operational purposes in these cases, and an external neutron source will still be employed.