References to the natural world are sprinkled all through Lady Chatterley's Lover, from the long, effusive passages about the beauty of the woods to little side comments, like Mrs. Bolton's thought that doctors could "sort of graft seed" to make Connie pregnant (11.10)—grafting being a gardening technique where you stick one plant onto another.
Or take Clifford's anger that Connie wants to hire a nurse, which he says kills "the real flower of the intimacy between him and her," although Connie thinks of that flower "like an orchid, a bulb stuck parasitic on her tree of life" (7.130). Flower, trees, rabbits, pheasants: Lady Chatterley's Lover is overrun with flora and fauna.
So what's all this natural imagery doing? Well, for one thing, it offers a nice contrast to Lawrence's opinions about the soulless mechanical world. Underneath the misery of Wragby, there's a whole bounty of nature waiting to spring to life.
And take the references to paganism that Lawrence drops. When Mellors comes in from the rainstorm with an armful of flowers, Connie looks at him "as if he were not quite human" (15.144)—as if, maybe, he's an incarnation of Pan. Pan?
In Greek mythology, Pan is the god of the wild. He's associated with all sorts of other mythologies, like the Horned God of Celtic tradition. Mellors actually refers to him explicitly, saying that peasants "should be alive and frisky, and acknowledge the great god Pan. He's the only god for the masses, forever" (19.165).
In other words, the common people should be worshipping pagan gods with, like, orgies and bonfires and piles of flowers rather than sitting in dark, depressing churches. All the flowers and tiny forest creatures are pointing the way back to a (probably mythic) past when people lived in harmony with the gods of nature.
It does sound nice, but still—we'll take the indoor plumbing.