I done similar for a freind.
Not sure of the make, think it was likely friedland, they made a hand held push button transmitter that you could walk round with, you could buy it seperate from the wireless chime/reciever.
It would need to match the frequency of your unit.
Inside it has two designated connections, normally open, that when shorted transmitted a signal to activate the wireless bell.
As the other poster said, you cannot just connect anything across the wired bellpush due to the voltage present
Inside your wired bell theres loads of room, were there is no batteries, so inside there i put a small normally open relay across the supply terminals that become live when the wired buttons pressed.
The coil needs to match the output voltage from your newlec transfomer, possibly 8 or 12 volt ac.
The handheld transmitter has slots to hang on the wall so i mounted it next to the wired bell box, but you can locate it nearby out the way, just longer cable between.
I then took a two core small cable, from the transmitter, 2 normally open terminals, into the bell box and onto the 2 normally open contacts on the small relay.
So now you press the wired button, wired bell ding dongs as normal, but as its pressed it closes the relay, in turn shorting the 2 designated contacts in the transmitter, transmitting the signal and operates the wireless, receiver/ bell
This was about 15 years ago and struggling to find the transmitter on the net, but its possible there still made or something similar.
The idea was you could take the button in the garden or shed and still call the maid or whoever if needbe, it had like two rubber hooks on it too so you could just hang it up wherever you liked.
The receiver chime box was portable too, and you just sat it on a table, or say in the shed, while the girlfriend sat on the sofa with the button transmitter.
You can do it by opening your wireless push, seing what the button activates, if it just shorts 2 points, then connect with small wire the same 2 points across your normally open relay, put the wireless push and the relay in the wired bell box out of sight