National digital identity verification systems have played a critical role in the effective distribution of goods and services, particularly, in developing countries. Due to the cost involved in deploying and maintaining such systems, combined with a lack of in-house technical expertise, governments seek to outsource this service to third-party cloud service providers to the extent possible. This leads to increased concerns regarding the privacy of users' personal data. In this work, we propose a practical privacy-preserving digital identity (ID) verification protocol where the third-party cloud services process the identity data encrypted using a (single-key) Fully Homomorphic Encryption (FHE) scheme such as BFV. Though the role of a trusted entity such as government is not completely eliminated, our protocol does significantly reduces the computation load on such parties. A challenge in implementing a privacy-preserving ID verification protocol using FHE is to support various types of queries such as exact and/or fuzzy demographic and biometric matches including secure age comparisons. From a cryptographic engineering perspective, our main technical contribution is a user data encoding scheme that encodes demographic and biometric user data in only two BFV ciphertexts and yet facilitates us to outsource various types of ID verification queries to a third-party cloud. Our encoding scheme also ensures that the only computation done by the trusted entity is a query-agnostic "extended" decryption. This is in stark contrast with recent works that outsource all the non-arithmetic operations to a trusted server. We implement our protocol using the Microsoft SEAL FHE library and demonstrate its practicality.