Does WiFi still work if I use my router as a switch?

I am buying a new router want to set up my current router in the living room to act as a switch and Wifi access point. See image below, where lines are wired connections.

I've read guides explaining that to use the router as a switch. I need to only use the LAN ports on the back and turn off DHCP and NAT gateway. What I have not found is, can the router still act as a Wifi access point?

I would rather have the Wifi access point from the living room than where the router currently sits.

Yes. By turning off dhcp and wan but leaving the WIFI on you are in effect turning it into an access point.

You will likely want to move its management IP to something other then the default, and make sure its in the same subnet as the new routers LAN range, and as you stated ensure DHCP is off.

If your new router includes WIFI, if you have same SSID and password but different channels you will also have almost seamless roaming between the two.

I have just done this in my house. I have a new router with a few devices connected (both wired and wireless), and I have my old router wired to it and set up as an access point, with additional devices connected to it (both wired and wireless).

Besides changing the settings on the old router to work as an access point, I also had to connect the cable coming from the new router to one of the four "outputs" on the old router, not to the "input" labelled "WAN". This seems counter-intuitive, but that's how things work if you use a router as an access point.

I also gave the access point a fixed IP address in the settings of the new router; I'm not sure whether this is strictly necessary, but then you can easily reach the access point's web interface to change its settings.