I think the engine here definitely refers to the goldfinch and her family. Before she arrives, the laburnum tree is quiet, "quite still," with nothing to indicate that there is anything alive inside it. However, when the goldfinch arrives, the tree seems to surge to life, but the "machine" which springs into action here is not the tree itself but simply the creatures living inside the tree, who have now been stimulated to chitter and "tremor" at the arrival of their mother. Hughes describes this as "the engine of her family," a gathering of small birds who the goldfinch is able to "stoke" as she would an engine by feeding the baby birds with the food she has gathered for them.
To a certain extent, the "engine" is confined within the body of the tree—the laburnum is the casing of the engine, as it were. However, the imagery of the poem paints the mother goldfinch as the stoker, feeding the engine of her chicks in the same way that an engineer might feed a fire running an engine. It is not the silent tree which needs her attentions, but only the living machine she has stowed inside it.