Can I replace a 1N4148 diode with a 1N4007?
Though both are just normal diodes, they are meant for totally different purposes. You should check the datasheet of both. 

IN4148 : http://www.nxp.com/documents/data_sheet/1N4148_1N4448.pdf

IN4007 : http://html.alldatasheet.com/html-pdf/329387/CHENG-YI/IN4007/383/1/IN4007.html

Especially for,


Maximum Forward current
Maximum Forward Voltage drop
Switching speed (applicable when data transfers involved, sometimes schottky diodes preferred for this)
Maximum Reverse voltage.


With respect to above important characteristics of diodes you can notice from the datasheets that both IN4007 &amp; IN4148 have no near match values.

However, for the purpose of these diodes' interchangeability in your case, the parameter to be considered is the Maximum forward voltage, which is 1.1V for IN4007 (at 1A current, which will not happen with your system). The link you shared says VCC of your play station can vary from 5V down to 3V. 

So, your play station should work fine with IN4007.