I spent some time yesterday leafing through Ralph Griffiths' amazing tome The Reign of Henry VI which I think holds the record for the thickest book in my collection. 968 pages in hardback, costing £25 as far back as 1981 - I must have been loaded in those days! (Well, it was before I got married.)
Anyway, it appears that after the death of Ralph Neville, York and Joan Beaufort lived in the King's Household. (The latter is perhaps the more surprising.) Also in the same household was the King's mother, Katherine of Valois. The Council ordered that all royal wards should live with the King, suitably attended at the King's expense. It must have been rather crowded.
After the death of Henry V, the following arrangements evolved, though they were not what had been ordered in Henry V's will. Bedford, Henry VI's elder surviving uncle, spent most of his time in France, and acted as Regent there. However, when he did come home to England he was pre-eminent there as well.
The second uncle, Humphrey of Gloucester, stayed mainly in England at the head of the Council, but his role as Protector was tightly circumscribed, much to his distaste. He spent much of his time falling out with his uncle, Bishop Beaufort - the pair of them seem to have cordially detested one another. This was the political element - Henry VI himself was under the care of the Duke of Exeter. (Thomas Beaufort, the youngest son of John of Gaunt and Katherine Swynford.) Exeter's role seems to have been fairly hands-off and mainly delegated to deputies.
One thing this government failed to do was keep order in England. I was surprised how much violence and feuding there was at this time - everyone seems to have been at it, not least John Talbot (later Earl of Shrewsbury) and Joan, Lady Abergavenny, who as important members of the nobility really ought to have known better.
In the era of Richard II domestic violence is often blamed on the absence of a decent war in France to keep the thugs busy. Obviously this argument (which I've accepted myself at times) is deeply flawed, as in the 1420s there was a fair old war going on in France and it clearly did not keep things quiet at home. Nor can Henry VI be blamed at this stage - he was a little boy, and not involved in government. It seems the English (and Welsh) were just a rowdy lot and enjoyed a bit of casual violence against their neighbours.