@Julian_R 

Hi, for your case, here are some suggestions. 
1. Please check whether Wireless password of your new router you entered is correct or not? Please go to Settings--WiFi on your phone to forget the new router WiFi, then reconnect it using the same Wi-Fi password to see if it works. 
2. Please reset the extender into default settings again by holding the reset button until all lights flashing once, then refer to this instruction to configure your range extender again carefully:

Via Tether App: https://www.tp-link.com/support/faq/1373/

Via Web interface: https://www.tp-link.com/support/faq/1400/

Note: During the configuration, please make sure that the extender is close to your router and be careful when you enter router's wireless password.  

If it still doesn't connect, please check if there is any special settings on your main router, like access control etc. which will block the extender.

If no such special settings, please let us know the following information:

1) the LED status of your range extender

2) distance between range extender and your router

3) model number and hardware version of your main router

  @Julian_R 

Hi, you mean only power LED on the extender is on, all the other LEDs are off, right? is there any error message when you configure the extender?

May I know if the 2.4G and 5G WiFi name are the same on your router? If so, could you please try to change the WiFi name of your router to use different names for the two bands? Then please reset the extender and configure it again to connect the new WiFi with correct password.

If still only power LED is on, please refer to the instruction here to check your router settings:

Recommended settings on the host router to work well with TP-Link Range Extenders

Hope it helps!