Vehicular Ad-hoc Networks (VANETs) constitute crucial elements within intelligent transportation systems. However, the rapid development of VANETs has brought forth an increasing number of security concerns. Conditional Privacy-Preserving Certificateless Signature (CPP-CLS) has emerged as a promising solution to ensure data security, preserve vehicle anonymity, and establish unlinkability in VANETs. In contrast to traditional public key infrastructure systems that involve cumbersome certificate management, and identity-based frameworks fraught with key escrow issues, CPP-CLS presents a more apt approach for VANETs. Unfortunately, the researches on CPP-CLS present a strange phenomenon in that a scheme proposed is always pointed out to have various security problems, especially public key replacement attacks. Moreover, there is a scarcity of published researches on the generic construction of CPP-CLS. To tackle these challenges, this paper proposes the first generic construction for CPP-CLS based on Type-T (Three-move type) signature, in which the public key reconstruction technique enables any receiver who owns a part of the sender's public key and the KGC's public key to reconstruct the complete sender's public key, which can alleviate the public key replacement attacks. A formal security analysis proves that our scheme effectively guards against existential forgery under adaptively chosen message attacks in the random oracle model, contingent upon the security of the underlying Type-T signature. Furthermore, We provide two specific instantiations of the generic construction to verify feasibility. Among them, the instantiation based on module learning with errors is effective against quantum attacks. Based on extensive experimental results and theoretical analysis, our implementations surpass the majority of existing similar schemes in either performance or security. This substantiates the feasibility of our generic scheme, making it applicable for constructing CPP-CLS schemes.