Let us assume you are trying to reach the NYTimes, www.nytimes.com.
Routing with NAT means that your router will send your packets on to its gateway (routing) while re-writing the source address (which originally is your IP address) to its own (WAN!) address (this rewriting is NAT). The reason for this is that your router's upstream router does not know of your existence, it only knows the IP address of the WAN interface of your gateway, so, when it will have to relay back the NYTimes' reply, it will be able to let you have it if and only if it can first reach your router/gateway. All along, the gateway keeps track of all packets sent, so that when it gets a reply from the NYT it knows it has to send it to you, not to your wife's pc.
This is how a gateway works: the gateway joins two distinct subnets.
You might have decided that you wanted to use your ISP-supplied gateway as the gateway for your whole LAN, providing routing, DHCP and DNS services. In this case, how would you have had to configure your routerB? Well, in this case its role would have been simply that of passing packets on to their intended recipients. This is the scenario under which you would want to have the Bridged configuration: packets would be routed and NATted by routerA, and routerB would just pass them along. The fact that routerB also acts as an AP (access point) changes nothing. Probably, this would be the fastest configuration because you would drop one layer of NATting, but I doubt that, nowadays, this would make much difference.
The third scenario, routing without NAT, comes in handy when, once again, the LAN is controlled by your routerA (including DHCP, DNS, and routing), but for some reason you want some of your packets not to be routed the traditional way, but thru some other complex scheme: for instance, if you wish to use some kind of proxy that makes it look like you are in some far and remote place. Or. alternatively, you might setup a proxy on routerB which filters out porn sites and Facebook, so as to protect your children.
In either case, you would want some pcs to go the usual route, while others should be going thru the proxy: then you have all of these machines routed by routerB thru the proxy. There is no need for the reply packets to be NATted, because the reply packets do not need to be acted upon by routerB, so they may just as well come to you directly without further delay. RouterB would be acting then as a pure router, not a gateway, because it is not separating two distinct subnets.
What would happen to your communication?
In the first case, if you were to pass routerB suddenly into
Bridged mode, for about an hour it would disrupt communications (because most machines still act with the old IP address, which however however does not have a router any more); this period ends when the
lease of the IP address ends, about 1 hour later for many LANs. Then all machines will ask for a renewal of the lease, but will find a new DHCP server which will give them new IP addresses. Then everything will start working again. You must remember, though, to disable the DHCP server on routerB: though this stands to reason, it is not necessarily automatic.
In the second case, the answer is exactly the same, whether you enable connection via a proxy or not.