If late is an adjective, then that would make boarding the plane an adjective complement (Cf. a meaning similar to ungrammatical *We were late to board the plane).
If late isn't an adjective, then it would be an adverb situated in the middle of a progressive construct (Cf. adverb placement in We were boarding the plane late. Here late modifies boarding, whereas in your example, late post-modifies the verb, We were late.)
- We were happily boarding the train. <adverb>
- We were happy boarding the train. <adjective>
- We were happy to board the train. <adjective>
- We were late boarding the train. <adverb>
- We were late to board the train.
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