I'd be very, very surprised if it got anything more than 1 kbps in the next couple of years. Very likely much less (think several seconds to send a single SMS-length message).
That's how much Iridium and Globalstar (the two existing comparable LEO solutions) can do without large external antennas for current satellite messengers (Garmin InReach, SPOT, more recently Apple iPhones etc.)
Actually using 1-2 Mbps will likely remain unaffordable for a while even once it becomes possible, given the cell sizes and frequencies involved.
Starlink is much cheaper than that because it uses the Ka-band (which has tens to hundreds of times more bandwidth available for this application) and directional beams on both ends of the connection, vastly increasing spatial reuse.
This was the number I was remembering, which was "two to four megabits per cell zone" so definitely not _per device_ (unless you are in a lonely cell).
For Iridium (which also uses LEO satellites and spot beams), the typical cell size is tens of thousands of square miles, though!
Even if Starlink manages to provide much smaller cells (which is not trivial, especially with omnidirectional mobile devices as clients instead of phased-array dishes), we're talking several orders of magnitudes compared to a regular terrestrial cell.
FWIW you can use Starlink with a DC power supply and under volt it to get pretty decent power draw.