Are the "Bride of Christ" in the New Testament and the Lord's description of Israel as His wife in the Old Testament speaking of the same entity?
The Lord assigns female identities to both Israel and to the Church, but they are not meant to be confused with one another. They are different identities.
The Lord refers to Israel as the wife of the Lord:
Is. 54:6 “For the LORD has called you,
Like a wife forsaken and grieved in spirit,
Even like a wife of one’s youth when she is rejected,”
Says your God.
On the other hand, the Church is called the Bride of Christ:
Rev. 19:7 “Let us rejoice and be glad and give the glory to Him, for the marriage of the Lamb has come and His bride has made herself ready.”
Though these two references are similar, they are not the same. In scripture the wife of the Lord is consistently a reference to the nation of Israel only. The Church is never called a "wife." On the other hand, the Bride of Christ (or just bride) is always used in reference to the (primarily Gentile) Church of the New Testament. These are the ones who have been called into faith since Pentecost and remain until the Church is removed from the earth at the Resurrection.
These two references reflect the two-part nature of God's plan of redemption for the world. Salvation comes to the Jew first, but the Gentiles are also given opportunity through their association with Israel and the promises made to that nation. Both groups are "wedded" to the Lord in that both are included in a single covenant given to Abraham and fulfilled in Christ.
Our Isaiah study provides a complete and detailed explanation on the relationship between Israel and the Church.