Mechanism for the conversion of an unsaturated alkyl halide to an alcohol
For what it's worth, this reaction does not appear in the literature. So, whether it actually forms the product or not is anybody's guess.

However, if I had to suggest a mechanism... I would note first that an SN1 reaction on a primary centre, as you have proposed, is difficult. I think it more likely that the alkene assists in the expulsion of the chloride.



The resulting cyclopropylmethyl carbocation is stabilised and is electrophilic at multiple sites. I have only drawn one resonance form, but there are a few more. Attack of water at the tertiary carbon leads to the desired product after deprotonation.