Failover of a LAG connection means that the traffic would failover to the link(s) still up. It is much, much faster than STP failover. It is even faster than routing protocol failover.
Your BFD and OSPF session would remain up if there is at least one link of the LAG still up.
Don't confuse the LAG interface, which OSPF and BFD use, with the individual interfaces of the LAG links. As long as the LAG interface is still up, and it will be if even one of the LAG links is still up, then OSPF will continue to run on the interface. The LAG interface decides which link a flow will use, and it will shift flows to remaining links when any link goes down. This is invisible to the flow. This happens very, very fast, and there is no way that OSPF would miss enough Hellos to decide the neighbor is down.