Assuming typical "soho" 4LAN + 1WAN type routers, you will have stability issues and confusion - the main stability issue will because you are running 2 DHCP servers - depending on the IP ranges this could mean 2 systems (or worse, 2 routers) get the same IP address, which will be problematic. Best case scenario, the link causes enough latency that each side prefers the closer DHCP server and thus uses it to route traffic - assuming the routers don't assign overlapping IP's.
Setting up automatic failure if 1 connection fails is HARD. There is no single simple solution. It would be quite hard, but (if I was on a budget), I'd make sure the routers spoke OpenWRT, and convert each router so it had 3 LAN ports and 2 WAN ports. The LAN ports for each router would feed the local building, with 1 WAN port feeding the WIFI bridge, and the second one feeding the Internet connection. You could then use some mechanism (depending on the connections) to balance / direct traffic either directly out the WAN to the Internet or out the WIFI bridge to the WAN of the other device. This is doable, but is quite advanced and requires a good understanding of routing I fear you do not have [ based on the question ].
You would probably be better off with a default configuration and some kind of manual fail over - eg everything goes through 1 router (with the second router off). If that connection fails turn off 1st router and turn on second router - being the simplest. In that case you want to make sure the routers have the same LAN gateway address, both run DHCP servers and that the IP addresses issued by DHCP don't overlap.