actually, the OP is correct, the reaction will probably proceed (and it will be Sn1). Using a polar aprotic solvent isn't exactly bad for Sn1, it's just not as good as a polar protic solvent. Technically polar aprotic solvents can also give stability to carbocations, just not as much as polar protic solvents. Given the fact that the alkyl halide is tertiary, the extra stability gained by using a polar protic solvent is not as important.
The nature of the nucleophile is irrelevant. For an Sn1 reaction, the nucleophile is not part of the rate determing step so its characteristics don't affect the rate of the reaction (unless you get to the point where it's such a bad nucleophile that it becomes rate-limiting.)
Of course, this also depends on the nature of the alkyl halide. If each of the three R groups is a trifluoromethyl group (the leaving group isn't fluorine, it's just of the formula (F3C)3CX where X is the leaving group), then you're not going to get an Sn1 reaction because the carbocation would be extremely electron deficient. But for a normal alkyl halide without electron withdrawing groups, the reaction should proceed.
So yeah, i know they teach that you use polar protic solvents for Sn1 and polar aprotic solvents for Sn2, but, in the case of Sn1, polar protic solvents are just better, but not required, especially if the reaction would proceed through a tertiary carbocation. Polar protic solvents add a bit more stability to the carbocation, but polar aprotic solvents don't decrease the stability, they just don't add as much to it (they do add to it though). For Sn2, polar aprotic solvents are more of a requirement (although there are cases where this can be overcome), because a polar protic solvent decreases the nucleophilicity of the nucleophile.
