The Church of England website has several prayers composed with regard to the current pandemic.
Many of these follow the typical Church of England practice with regard to the direction of prayer. That is they are addressed to God the Father through Jesus Christ.
Typically Church of England prayers begin by invoking God, often mentioning some attribute, such as Merciful God, Almighty God etc. It is not at this point immediately clear that it is God the Father, rather than the Holy Trinity, who is being addressed, but the end of the prayer is usually a variation on "through Jesus Christ our Lord". Also, during the prayer, Jesus and the Holy Spirit may be referred to in the third person.
The collect for Easter Day goes:
Almighty God, who through thine only-begotten Son Jesus Christ has overcome death, and opened to us the gate of everlasting life; We humbly beseech thee, that, as by thy special grace preventing us thou dost put into our minds good desires, so by thy continual help we may bring the same to good effect; through Jesus Christ our Lord, who liveth and reigneth with thee and the Holy Ghost, ever one God, world without end. Amen.
Here the addressee is "Almighty God" but Jesus is referred to as "thine only-begotten Son" making it clear that we are talking particularly to God the Father. The prayer is "through Jesus Christ" and again He is said to "live and reign with Thee and the Holy Ghost".
However the website also has a prayer addressed to Jesus.
The Litany in the Prayer Book contains these lines:
O God the Father of heaven: have mercy upon us miserable sinners.
O God the Son, Redeemer of the world: have mercy upon us miserable sinners.
O God the Holy Ghost, proceeding from the Father and the Son: have mercy upon us miserable sinners.
O holy, blessed and glorious Trinity, three Persons and one God: have mercy upon us miserable sinners.
Prayer, in the Church of England, can be to any of the Divine Persons, God the Father, God the Son or God the Holy Spirit, or it can be to the Trinity.
Article XXII of the Church's doctrine is:
The Romish Doctrine concerning Purgatory, Pardons, Worshipping and Adoration, as well of Images as of Reliques, and also the invocation of Saints, is a fond thing vainly invented, and grounded upon no warranty of Scripture, but rather repugmnant to the Word of God.
Regarding the invocation of saints as it does, the Church of England does not endorse the practice of directing prayers to or through the saints. Some members do it though.
Generally members of the Church of England will understand how the Queen expects them to pray, and how she herself will be praying. But Her Majesty wears many hats, or rather her hat, the Crown, symbolises many things. As well as being Supreme Governor (colloquially "Head") of the Church of England she is an ordinary, and committed, member of the Church of Scotland. She attends from time to time the General Synod in England and the General Assembly in Scotland. This year Prince William was to be her personal representative, or High Commissioner, until the pandemic forced the cancellation of the Kirk's General Assembly.
But she is also Queen of sixteen independent nations, and Head of the Commonwealth of 53 countries, some predominantly of a non-Christian religion. She is also, by longevity if nothing else, the most senior world leader and may be heard across the globe. She speaks not only as a religious leader but also in a wider capacity. When she calls upon those of all faiths and none to join in prayer and meditation, she will know that for convinced atheists, meditation rather than prayer will be their starting point. She will know that non-Christians will pray as they will. And she will hope that Christians of all denominations and in all countries, will join her in praying to God in whatever way they will.
But for the Church of England, the direction of prayer is to any of the Divine Persons, or to the Holy Trinity; but typically to God the father through God the Son.