To the best of my knowledge, the UK Home Office & UKV&I do not share information directly with Citizenship and Immigration Canada (CIC). They can not apply for a SAR in your name, only you can.
But they can require you to apply for a SAR and submit it to them. I know that a British colleague of mine who migrated to Canada had to get his criminal records from ACPO and submit that to the Canadian authorities as a part of his Canadian permanent immigration application.
So, if they need to contact the Home Office, it would be through you.
Be aware that the intelligence services of the five major Anglophone countries (US, UK, Canada, Australia and New Zealand), the so-called
Five Eyes, do share intelligence data. So if you are lucky enough to warrant intelligence surveillance, that information will be shared across the intelligence agencies. But that will not be CIC directly asking for your information from the Home Office.
I am not a lawyer or immigration advisor. My statements/comments do not constitute legal advice. E&OE. Please do not PM me for advice.