Hello Yash
Remember that the BDR forms adjacencies with all other routers in the network segment including the DR. That means that all updates exchanged between the DR and DROTHER routers in a particular network segment are also simultaneously received by the BDR. So at any given time the BDR is completely up to date.
Now if the DR fails, it will take the same amount of time for the network to reconverge as it takes for the adjacency to be detected. If the DR and BDR routers are directly connected and the interface goes down, this will be detected immediately since the BDR will detect the interface going down and it will immediately become the DR. If they are connected via a switch, then the detection will take longer, on the order of the hello and dead intervals.
But even if a DR goes down, the network will still be functioning assuming it is a stable OSPF topology. If the network is stable during the whole duration of the detection of the DR going down, hosts should not realize that any failure has occurred (except for routes that may require the use of the failed DR router to be routed)
If during that time between the DR going down and the BDR detecting it there is a change in the network, then convergence may take longer.
In order to speed convergence up, you can always use Bidirectional Forwarding Detection (BFD), a super-fast protocol used to detect failed router adjacencies in milliseconds or less! You can find out more about that here:
I hope this has been helpful!
Laz