「待望の赤ちゃん」は long-awaited baby と言えます。
例:
My brother and his wife were married for 10 years before they had their long-awaited baby boy last year.
「弟と弟の奥さんは結婚して10年になるが、昨年待望の男の赤ちゃんを授かった。」
had one's long-awaited baby で「待望の赤ちゃんを授かる」と言えます。
We're finally going to have the baby we've been waiting for for 5 years.
「5年間待ち望んでいた赤ちゃんをやっと授かった。」
baby we've been waiting for を使っても「待望の赤ちゃん」を表現できますよ。
ご参考まで!