Here is a quick topology of his network design:
At high level the design looks pretty simple. R1 & R2 are peering with WAN cloud using E-BGP. Between R1-R2-R3-R4-R5 we are running OSPF Area 0. To ensure R3-R4-R5 have reachability to external prefixes (Since it's always good to avoid BGP to IGP Redistribution), R1 & R2 are injecting default route into the network using "default-information originate always metric-type 1" option.
Also R1 & R2 are connected to each other over a link running OSPF.
Now let's talk about problem...which is the fun part :)
If R1's WAN connection goes down, R4 can't reach any external prefixes using default route any longer. Whereas at the same time both R3 & R5 don't lose their connectivity to same external prefixes.
Now ideally once R1 find it's WAN interface down, it should get default route from R2 and should be able to route R4's traffic in that direction.
Let me show you same behaviour using CLI :