Code-switching is a widely prevalent linguistic phenomenon in multilingual societies like India. Building speech-to-text models for code-switched speech is challenging due to limited availability of datasets. In this work, we focus on the problem of spoken translation (ST) of code-switched speech in Indian languages to English text. We present a new end-to-end model architecture COSTA that scaffolds on pretrained automatic speech recognition (ASR) and machine translation (MT) modules (that are more widely available for many languages). Speech and ASR text representations are fused using an aligned interleaving scheme and are fed further as input to a pretrained MT module; the whole pipeline is then trained end-to-end for spoken translation using synthetically created ST data. We also release a new evaluation benchmark for code-switched Bengali-English, Hindi-English, Marathi-English and Telugu- English speech to English text. COSTA significantly outperforms many competitive cascaded and end-to-end multimodal baselines by up to 3.5 BLEU points.