It might be as simple as the kickdown cable out of adjustment (too tight).

Check the power switch. It is on the facing of the transmission shifter,
left hand side,  maybe 10 o'clock position. It should not be pressed
down. If it is, shifts will occur at higher RPM than normal.

Also - longer shot - check the overdrive button. On my 4cyl 1995 camry,
the overdrive switch is on the gear shift about where your right thumb
would go. Make sure it is pressed in. This really should only affect
higher speed riding.

HTH,
Russell

Draining the transfluid and replacing with new --- and do again at next oil
change (and maybe even the next)  might help too.

You can check the trans by disconnecting the solenoid connector into
the tranny and try shifting manually from 1st to 2nd with the shifter
as you accelerate.  If it shifts ok like that, then it may be something
like a bad solenoid or speed sensor, but then you should also see a
code being thrown.  If it doesn't and you still seem to get a late
shift putting it into 2nd manually with the shifter, then your
transmission is probably going and will only get worse quickly the
longer you drive it like this.  As already suggested, you could try a
fresh fluid drain and fill and make sure the fluid level is ok (it can
be tricky to read the trans dip stick sometimes and you usually need to
take several readings before and after driving the car).  I think your
transmission may have the PWM pressure solenoid too which may be a
cause.  Before anything, do check for any thrown codes with am OBDII
scanner.

cheers,

davemc