How did Shakespeare celebrate the royal birth?

In the frenzy of royal birth and among all the news the one I like best is the royal crier proclaiming the birth of the royal baby outside St. Mary’s Hospital, accompanied by all the pomp and circumstance of  “this sceptered Isle” that boasts Shakespeare.

It would have been much better had the Archbishop of Canterbury visited just like Thomas Cranmer (the Archbishop of Canterbury during Henry VIII’s reign) did when Queen Elizabeth I (the daughter of Henry VIII and Ann Boleyn) was born.

We read this in Act V Scene 5 of Henry VIII,

CRANMER

[Kneeling] And to your royal grace, and the good queen,
My noble partners, and myself, thus pray:
All comfort, joy, in this most gracious lady,
Heaven ever laid up to make parents happy,
May hourly fall upon ye!

KING HENRY VIII kisses the child
With this kiss take my blessing: God protect thee!
Into whose hand I give thy life.

CRANMER
Let me speak, sir,
For heaven now bids me; and the words I utter
Let none think flattery, for they'll find 'em truth.
This royal infant--heaven still move about her!--
Though in her cradle, yet now promises
Upon this land a thousand thousand blessings,
Which time shall bring to ripeness: she shall be--
But few now living can behold that goodness--
A pattern to all princes living with her,
And all that shall succeed: Saba was never
More covetous of wisdom and fair virtue
Than this pure soul shall be: all princely graces,
That mould up such a mighty piece as this is,
With all the virtues that attend the good,
Shall still be doubled on her: truth shall nurse her,
Holy and heavenly thoughts still counsel her:
She shall be loved and fear'd: her own shall bless her;
Her foes shake like a field of beaten corn,
And hang their heads with sorrow: good grows with her:
In her days every man shall eat in safety,
Under his own vine, what he plants; and sing
The merry songs of peace to all his neighbours:
God shall be truly known; and those about her
From her shall read the perfect ways of honour,
And by those claim their greatness, not by blood.
Nor shall this peace sleep with her: but as when
The bird of wonder dies, the maiden phoenix,
Her ashes new create another heir,
As great in admiration as herself;
So shall she leave her blessedness to one,
When heaven shall call her from this cloud of darkness,
Who from the sacred ashes of her honour
Shall star-like rise, as great in fame as she was,
And so stand fix'd: peace, plenty, love, truth, terror,
That were the servants to this chosen infant,
Shall then be his, and like a vine grow to him:
Wherever the bright sun of heaven shall shine,
His honour and the greatness of his name
Shall be, and make new nations: he shall flourish,
And, like a mountain cedar, reach his branches
To all the plains about him: our children's children
Shall see this, and bless heaven.

KING HENRY VIII
Thou speakest wonders