Here are some guide questions:
1. If changing hosting providers, will the website content stay the same, if there's one?
2. Is it easy changing nameservers with your current registrar?
3. If yes to no. 2, how soon will the registrar submit the change, batches or real time?
That way, while the registrar transfer occurs, ideally there won't be downtime. And you
can't make any changes while the transfer is taking place.
Following a change of Registrar, the domain record is locked for a period of time and during this, nameserver details can't be updated.
Changing hosts first means that nameserver details can be updated at your existing Registrar then, following the changeover, the intra-Registrar transfer request can be initiated.
As a point of note, the lock period can be up to 14 days, but seems to be at the Registrar's discretion, so is often less than this.
R.