It varies; for direct peering/connections where a link goes down (and fast failover protocol is used), usually 5 seconds or so. Global propagation takes in the order of 5 minutes or less: https://labs.ripe.net/author/vastur/the-shape-of-a-bgp-updat...

They aren't really cached. Routes are kept in the routing table as long as peers are responsive, if they stop responding to hello's or a link goes down to directly connected peer the routes are removed (or replaced with a less preferred one). While its possible to configure a router to keep routes in the table for a peer that is down for some time its typically not used unless you have a good reason for it.

They are added and withdrawn on demand, or dropped all at once if router loses connectivity with another.

So really it's processing time of every node in the chain. Few seconds to few minutes, not really that different from DNS.

The advantage is that there is no updates if there is no changes, instead of update every time TTL expires but that's about it.

routing tables usually aren't "cached" in the same sense that DNS caches entries.

however. Inside ISP;s routes are usually distributed by an IBGP with a couple of route reflectors, and route reflectors will drop routes if no router within the IBGP domain is advertising the route.

Routes between ISP's are usually fairly static thanks to mechanisms like graceful restart, which makes sure routes do not get dropped, even if BGP has a restart between routers. THere are various other mechanism in which to achieve this. (look up route flap dampening if you want to learn more).