This paper investigates four types of cross-utterance speech contexts modeling approaches for streaming and non-streaming Conformer-Transformer (C-T) ASR systems: i) input audio feature concatenation; ii) cross-utterance Encoder embedding concatenation; iii) cross-utterance Encoder embedding pooling projection; or iv) a novel chunk-based approach applied to C-T models for the first time. An efficient batch-training scheme is proposed for contextual C-Ts that uses spliced speech utterances within each minibatch to minimize the synchronization overhead while preserving the sequential order of cross-utterance speech contexts. Experiments are conducted on four benchmark speech datasets across three languages: the English GigaSpeech and Mandarin Wenetspeech corpora used in contextual C-T models pre-training; and the English DementiaBank Pitt and Cantonese JCCOCC MoCA elderly speech datasets used in domain fine-tuning. The best performing contextual C-T systems consistently outperform their respective baselines using no cross-utterance speech contexts in pre-training and fine-tuning stages with statistically significant average word error rate (WER) or character error rate (CER) reductions up to 0.9%, 1.1%, 0.51%, and 0.98% absolute (6.0%, 5.4%, 2.0%, and 3.4% relative) on the four tasks respectively. Their performance competitiveness against Wav2vec2.0-Conformer, XLSR-128, and Whisper models highlights the potential benefit of incorporating cross-utterance speech contexts into current speech foundation models.