Can I complete the theoretical and practical elements of aircraft type training at different Part-147 organisations?

Yes. The theoretical and practical elements of aircraft type training do not have to be completed by the same organisation. Each element must independently comply with Appendix III to Part-66 and be conducted by a maintenance training organisation appropriately approved under Part-147 or, where conducted by another organisation, be directly approved by the competent authority in accordance with point 66.B.130. Reference: Appendix III to Part-66 and point 66.B.130 of Annex III (Part-66).