In the same year of 1562 Queen Elizabeth became infected with smallpox. This nearly killed her and left her with pock marks that she covered with the white makeup that was popular at the time.
The Doctor or your PCs might be around to help save the life of Queen Elizabeth. The smallpox could turn out to be an alien disease, perhaps contracted from the Zygons or other aliens.
This brush with death caused her advisers to suggest that she marry and produce heirs as soon as possible. While she courted many and arranged marriage negotiations with foreign royals she never did marry.
We can assume that Elizabeth kept her marriage to the Doctor secret and while she saw the advantage of opening diplomatic channels through the pretence of arranging marriage she honoured her bond to him.
You might want to examine her relationships in closer detail. After so many years she may have given up hope and be about to marry someone else when the Doctor (or other Time Lord PC) arrives and reminds her of her vows.
If the Doctor truly did sleep with Queen Elizabeth is is possible that she had his child? If so it would mean that his child would have a claim to the throne, not only threatening history but placing the child in danger from those who wish to claim rulership of the country.
Queen Elizabeth, in time of conflict, did her best to broker peace. This befitting someone chosen by the Doctor. Did he influence her rulership or is this a characteristic that she already possessed which attracted the Doctor?
PCs in this period know that they can rely on Queen Elizabeth to seek a peaceful outcome. As long as she can be persuaded that her kingdom will not be endangered she can be a powerful ally.
A constant threat to Queen Elizabeth came from her half-sister Mary, Queen of Scots and the Catholics who supported her. While it is easy to paint them as villains they did legitimately believe that Queen Elizabeth, according to their religion, had no right to be on the throne.
This led to such events as the revolt of the Northern Earls in 1569, which received support from Pope Pius V. This created an atmosphere of mistrust where any Catholic was a potential traitor, with measure coming into place by 1570 to exclude them from power.
Allies of Queen Elizabeth have plenty of opportunities to foil plots to remove her from the throne. They can do this out of loyalty, love or just to ensure that history is protected. Aliens and time travellers might support the rebels in order to seize the country for themselves.
Persecution is something that Doctor Who has explored before and the PCs might find themselves protecting Catholics. While they can’t change society they can still make sure that innocent lives are not lost.
Queen Elizabeth was still not safe, even after the death of her half-sister. Philip II of Spain made a claim that his daughter, Isabella, had a claim to the throne as she was descended from Edward II. To this end he sent an Armada to invade England in 1588.
This gives the PCs an opportunity to be involved in a historic conflict and meet such prominent figures as Sir Francis Drake. It is notable that part of the reason the invasion failed was due to unusually strong storms and cold weather. The PCs might be responsible for this or they can seek out the source (which might be alien in nature).
Due to the fact that the Spanish had used Ireland as a base Queen Elizabeth declared war on the island, to bring the country under control. This lasted between 1594 and 1603, with the high cost of the campaign resulting in an economic crisis between 1594 and 1597 not to mention further provoking outrage from Nationalists and Catholics.
These events allow you to have adventures based in Ireland during the war, a dangerous but pivotal time. While the United Kingdom has frequently been explored in Doctor Who there are plenty of secrets that you could have unearthed during the conflict.
The resulting high taxes and famine in Britain can be stark reminder of the price of conflict. How would the Doctor react to seeing what his Elizabeth had done? Such is the problem with relationships with historical figures. You can’t change their behaviour without altering time.
‘The Shakespeare Code’ takes place in 1599, 37 years after the events of ‘The Day Of The Doctor’. We know at this point she considers the Doctor her sworn enemy.
Is this because of his abandonment or did something else occur? It could be that the Doctor did oppose her or carry out some other act that made her so angry. At the time the Doctor didn’t bother to investigate but now that he knows what their relationship is he might investigate.
This can be the basis for an adventure for the Doctor or another group of time travellers. It could be that the very action of investigating puts in motion events that result in Queen Elizabeth considering the Doctor her enemy.
If you want to preserve the relationship between them then it could be that the Queen Elizabeth in ‘The Shakespeare Code’ is another imposter. The Doctor might realise that his Elizabeth would never act in this way and seek out the real Queen.
PCs who meet Queen Elizabeth in her early years could receive her patronage. Just as the Doctor was made Curator of the Under Gallery she might give them equally important responsibilities.
They could similarly receive sealed messages from the past, requesting they act to protect the country. This can be very puzzling for them if they haven’t yet met the Queen (or at least don’t remember it).