Move email/domain hosting to google or MS

Hello you beautiful IT peolple!
I don't know who will answer this post but you know more about the IT than I do :)
My website is hosted by 123-reg and develped on their website builder. The emails are also hosted by 123-reg.
I need the emails to be moved to googlmail or Microsoft servers- even if the domain stays with 123-reg.

Purpose: This is to improve the security of emails and deliverability of emails (then end up in junk)
I already pay for MS365 Office

On another domain, I have a similar deliverability and security issue.
It is hosted by Mzone and built-in a custom CSM. Not sure how easy it would be to move the hosting back to 123-reg and have the emails hosted by google or MS servers

Budget: Please provide the scope of work and estimated budget as well as what you need from me.
If you see risks in delivery of this project do highlight them at the outset.

This applies to any fixed or per-hour quotes you'd like to make.
Skillset is key. Cheaper is not necessarily better if you will take ages to deliver.

I'm based in London.

Regards
Eliza