Bitcoin takes into account clock of nodes on the network when propagating transactions tru the mempool and persisting them into the next block. If majority of the network clocks are compromised what kind of issues would occur?
How dependent is Bitcoin on the clock?
A lot less than many other applications.
A timestamp is accepted as valid if it is greater than the median timestamp of previous 11 blocks, and less than the network-adjusted time + 2 hours.
AFAIK this means that a block can have a timestamp that is earlier than that of its parent (a block can appear older than its parent). I vaguely recall that there are some such blocks in the Bitcoin blockchain.
So if computers become unable to keep good time to within 2 hours of the correct time, they will start to be problems, blocks will be rejected and the Bitcoin network might, I suppose, fragment.
Note that time synchronisation protocols such as NTP can synchronise times on computers to less than a few tens of milliseconds across the planet, despite significant latency and jitter. There's no good reason for computers to have time drifting apart by seconds, let alone hours.
Compromising clocks might be exploitable but it seems non-obvious.