Recent state-of-the-art neural text-to-speech (TTS) synthesis models have dramatically improved intelligibility and naturalness of generated speech from text. However, building a good bilingual or code-switched TTS for a particular voice is still a challenge. The main reason is that it is not easy to obtain a bilingual corpus from a speaker who achieves native-level fluency in both languages. In this paper, we explore the use of Mandarin speech recordings from a Mandarin speaker, and English speech recordings from another English speaker to build high-quality bilingual and code-switched TTS for both speakers. A Tacotron2-based cross-lingual voice conversion system is employed to generate the Mandarin speakers English speech and the English speakers Mandarin speech, which show good naturalness and speaker similarity. The obtained bilingual data are then augmented with code-switched utterances synthesized using a Transformer model. With these data, three neural TTS models -- Tacotron2, Transformer and FastSpeech are applied for building bilingual and code-switched TTS. Subjective evaluation results show that all the three systems can produce (near-)native-level speech in both languages for each of the speaker.