It's kind'a hard to see where the traces are going on the photos. One of the test points doesn't seem to correspond to either main antenna.
For each antenna one of the spring contacts should be ground and the other signal. The ground one should be connected to a large ground plane. The other to a relatively thin trace.
You would strip 1/4-1/2 inch of the outer insulation off the coax, undo the braid or foil and bundle it to one side, then strip the inner insulation from the center conductor. Use an xacto knife to carefully remove the green stuff from 1/8-1/4 inch of trace and a slightly bigger area of the ground plane directly adjoining.
Then solder the center conductor to the trace and the braid to the ground plane. Be particularly careful to work quickly with the center conductor because the insulation around it melts fast. Pre-tin the conductor. Pre-tin the PCB. Use lead-base solder if available (lower melting temp). Use lots of flux and a relatively low tip temperature. If it didn't work, cut an inch of coax off and start over
.
The main risk you have is pulling the trace up from the PCB. You can do that either by heating it up for a long time or by pulling on a wire that it soldered to it. SO avoid these things
. This also means that it may be smart to glue the coax to the PCB with some hot glue about an inch from the solder work to act as strain relief.
Edit: the ground plane sucks up a lot of heat, so you may want to heat it up 'til the solder melts and then dip the pre-tinned braid into the puddle.
