The big problem that I see with this arrangement is that the UK considers it work -- and it is; you will literally be working for compensation -- and will not allow you to enter for that purpose without a work visa.
If you have status as a long-term resident in Estonia, which you would normally acquire after five years living there, you can move to all but three other EU countries under the directive 2003/109/EC. See the European Commission's webpage for a brief discussion.
The three countries that have opted out of this directive are the United Kingdom, Ireland, and ...
Article 23 of Directive 2004/38/EC (the Free Movement Directive) allows the family members of EEA nationals who have the right of residence in the UK to work or become self-employed. https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/779784/free-movement-rights-direct-family-members-of-EEA-nationals-v7.0ext.pdf
A UK residence permit is required to apply in the UK for a visa for Estonia https://london.mfa.ee/how-to-apply-for-a-schengen-visa/ so that avenue is not going to work for you.
According to https://workinestonia.com/internationalhouse/how-to-get-documents/ it looks possible to apply for a residence permit after returning to Estonia on a C type visa, which ...
Your new child should apply for a tier 2 dependent visa:
Apply as a child
You were born in the UK
You’ll get the same permission to stay as your parent if you were born in the UK.
If you’re under 18
You can either:
be added to your parent’s next application as a dependant
To apply separately, ...