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Self-supervised-learning-based pre-trained models for speech data, such as Wav2Vec 2.0 (W2V2), have become the backbone of many speech tasks. In this paper, to achieve speaker diarisation and speech recognition using a single model, a tandem multitask training (TMT) method is proposed to fine-tune W2V2. For speaker diarisation, the tasks of voice activity detection (VAD) and speaker classification (SC) are required, and connectionist temporal classification (CTC) is used for ASR. The multitask framework implements VAD, SC, and ASR using an early layer, middle layer, and late layer of W2V2, which coincides with the order of segmenting the audio with VAD, clustering the segments based on speaker embeddings, and transcribing each segment with ASR. Experimental results on the augmented multi-party (AMI) dataset showed that using different W2V2 layers for VAD, SC, and ASR from the earlier to later layers for TMT not only saves computational cost, but also reduces diarisation error rates (DERs). Joint fine-tuning of VAD, SC, and ASR yielded 16%/17% relative reductions of DER with manual/automatic segmentation respectively, and consistent reductions in speaker attributed word error rate, compared to the baseline with separately fine-tuned models.