Sea levels have been rising "Funafuti Atoll, in the tropical Pacific Ocean, has experienced some of the highest rates of sea-level rise (∼5.1 ± 0.7 mm/yr), totaling ∼0.30 ± 0.04 m over the past 60 yr."
But the island is not sinking "Despite the magnitude of this rise, no islands have been lost, the majority have enlarged, and there has been a 7.3% increase in net island area over the past century (A.D. 1897–2013)."
Tuvalu is not sinking, it's actually growing in size. Although the shape of its coastline is changing which can displace some people. Linked in the BBC article you cite is this article (https://www.newscientist.com/article/dn27639-small-atoll-islands-may-grow-not-sink-as-sea-levels-rise/?ignored=irrelevant#.VX4fFGB3uCQ) which references this satellite imagine paper in Geology (https://pubs.geoscienceworld.org/gsa/geology/article-abstract/43/6/515/131899/Coral-islands-defy-sea-level-rise-over-the-past?redirectedFrom=fulltext)
Sea levels have been rising "Funafuti Atoll, in the tropical Pacific Ocean, has experienced some of the highest rates of sea-level rise (∼5.1 ± 0.7 mm/yr), totaling ∼0.30 ± 0.04 m over the past 60 yr."
But the island is not sinking "Despite the magnitude of this rise, no islands have been lost, the majority have enlarged, and there has been a 7.3% increase in net island area over the past century (A.D. 1897–2013)."