This is one of those phrases that holds cultural significance within a specific cultural group. That is likely why a word-for-word translation is not possible as the meaning will be affected. It is important to understand the cultural significance of sansankudo and that this ritual likely does not exist anywhere else in the world. One can, however, use the romanised phrase of "sansankudo" in conversation as it is unique to Japanese culture.
The closest equivalent saying would be "two become one" and is commonly used in Christian ceremonies. It simply means that the bride and groom will share their lives from now on and their actions will have an effect on their partner. In some Christian wedding ceremonies, the bride and groom may take a communion (drinking of wine and eating of bread that symbolise the blood and body of Christ, respectively) together, but there exists no specific saying relating to this.
"The bride and groom will now exchange vows" relates to the ritual of the couple each taking a turn to express the importance of the marriage to themselves. This is not a saying, but the phrase is commonly used in English ceremonies.
tetsuwan28さん
ご質問どうもありがとうございます。
上の文脈を英語で表現したい場合は、下記の言い方ではいかがでしょうか。
日本の場面を形容したい場合は、そのまま「三三九度」でもよいのではないかと思います。
例: Then the "Sansankudo" followed. The bride and groom exchanged cups of sake which is a Japanese tradition meaning they now have become one.
西洋人が伝わりやすい表現なら、「exchange vows」でもよいかもしれません。
ご参考にしていただければ幸いです。