What is the lifetime of a client's Wireguard public key?

The only way to make this system completely anonymous is if a separate Wireguard pubkey is used for each internet endpoint. E.g. if I visit stacker.news and dunlaoghairechessclub.ie and alcoholicsanonymous.ie at the same time, they should each go through separate Wireguard connections with separate, short-lived Wireguard pubkeys (see Tor's stream isolation concept). Otherwise the exit node can build a profile of the user from the set of visited sites and track them even across Wireguard pubkey changes. Most people have a quite distinct set of sites that they tend to visit.

The Wireguard protocol has a nanosecond-precision timestamp field which the exit node can use to perform identity correlation of clients based on their clock offset, drift and wander. Is anything done to address this?

The Wireguard implementation, at least on Linux, has a very weak mitigation against this issue in which the precision is reduced to 16.777216 ms, but a mere precision reduction is not enough to fully address this issue. As noted in the Wireguard paper, it must only be a per-peer monotonically increasing number. Ideally the field should be used as a counter starting at zero whenever a new Wireguard pubkey is used. Alternatively, assuming short-lived Wireguard pubkeys, one could use timestamps with per-pubkey random offsets.