There are various factors that contribute.
SN1 needs high temperature and a nucleophile, but the concentration of nucleophile doesn't matter. For the SN1 to occur, the leaving group has to dissociate and form a carbocation. If the temperature is too low, the SN1 wont occur.
In competition between nucleophilic substitution and elimination, higher heat tends to favor elimination over substitution. Substitution always takes 2 particles in (nucleophile and substrate) and generates 2 particles (leaving group and product), while elimination takes in 2 particles (base and substrate) and generates 3 particles (leaving group, protonated base, product). Increasing number of particles means increased entropy, and is favored by TΔS.
Elimination also needs high temperature and a base. Many nucleophiles are also bases (like hydroxide). E2 can be increased in rate by increasing concentration of base.
So in your original statement, you have tertiary alkyl halide, hydroxide and "no heat". Both SN1 and E2 will be slow with no heat. Which one will be favored depends on whether "no heat" is 40 C, 20 C, 0 C or -20 C. The lower the temperature, the more lilley the answer will be "no reaction". If "no heat" is 30-50 C, E2 will probably be more likely. Setting fluorides aside, Chlorides will favor E2 the most while Iodides will more easily dissociate and give SN1.
If you want pure E2, high concentration of hydroxide and about 80 C with the chloride or bromide would be good. This is not quite high enough temperature to get much dissociation and SN1.
If you want pure SN1, water instead of hydroxide and about 100 C would be good, probably with the bromide - that is enough heat to dissociate, and the water is a poor base. If you want to do SN1 at low Temperature, use an iodide - although tertiary iodides dissociate so easily they are not very stable.
There is a good explanation of this in the Organic review book by Fredlos.