Hi Vicky
Yes, you will completely destroy all local sites and the main site if you get this wrong.
1. The local sites should all be subdirectories of the main site so, for example, website.com/uk, website/com/fr etc.
2. Each site should be a unique entity, all support pages should be duplicated locally as well as listing of all products and services (local and global), with no links back to the mother site apart from maybe one in the 'about us' to show that there is a link between the two sites.
3. Add hreflang directives on every site page to its other versions (all of them) and an hreflang="x-default" to the main site. You need to do this so that Google does not consider each one a duplicate of the other.
https://moz.com/learn/seo/hreflang-tag
https://support.google.com/webmasters/answer/189077?hl=en
These will specify the location and language of the local sites.
The proposal that there will be multiple links back to the mother site will create an SEO nightmare - just don't do it. (Also, Imagine a basket with multiple items in, some from a local site and others from the mother site! - nightmare)
I hope that helps
Regards Nigel