If you wished to specifically follow the diagram in the App Note then a resistor from 1 kOhm or slightly higher value should work.
The 74HC645 is an octal transceiver chip with tri-state (3-state) outputs incorporating a direction input and output enable pin that may allow you to omit some of the other logic gates (ie the AND and Inverter gates).
It should also be possible if using bit-bashing for the SPI comms to consider a direct connection between the PICAXE and DS1620 temp sensor.
For a M2 or X2 PICAXE part, using a bidirectional IO, when set as an input it is in effect a tri-state output in the high impedance state, so switching to input mode when not specifically sending data to the DS1680.
For a M or X1 part where separate input and outputs are required, the PICAXE input can be wired directly to the DS1620 data pin and use a 1N914, 1N4148 or similar signal diode between the PICAXE output and the DS1620 data pin. The didode Kathode (banded end) towards the PICAXE output. It may be possible to use a resistor of around 1 k Ohm in lieu of the diode.