which is time needed to have the AD9635 up and running after power supply has been applied?
I'm not sure I can consider the value indicated for the "Wake-Up Time (Power-Down)" stated in the datasheet.
Yes, you are correct. The 375us wake-up time from power down is the time to wake-up from the power-down mode, and not from initially bringing up the power supplies.
When first powered up (and also after a Register 0x08 Digital Reset), the AD9635 goes through an initialization sequence. This sequence takes about 3 million sample clock cycles. This process starts after the analog is up.
At a sample clock frequency of 125MHz, 3 million clock cycles takes about 24ms. If we say that the analog take about 1ms to come up (375us + fudge factor), this gives us 25ms. So, conservatively let's say that 40ms is needed.
A lower sample rate will obviously increase the time required.