I think that we can assume that the initial hours would have been spent in shock and damage limitation.The heart of Westminster would have been strewn with dead bodies and, presumably, friends and relatives desperately searching for confirmation that their loved ones were amongst them.
But radiating out from Westminster Palace, many people would be injured and/or trapped. Flying debris and glass would ensure it. They would all need tending or removing to safety.
Meanwhile, an inferno was still raging and that would have to be dealt with too. With no organized fire brigades, this would need to be a co-operative venture amongst volunteers. Luckily, Westminster Palace is right next to the River Thames, so there would be no shortage of water.
In 1666, a human chain of people snaked from the river bank to wherever the fires had spread. They passed buckets and bowls of water up towards the flames. There's no reason why this scene wouldn't have happened in 1605 too. Unfortunately buckets of water aren't too effective when whole buildings are ablaze.
A frenzied hive of activity would be playing out down wind of the flames. Westminster was a center of trade and industry, so warehouses would be endangered, as well as family homes. All that could be saved would be loaded onto carts and hurried away.
Naturally that would lead to the narrow streets becoming blocked and travel times prolonged accordingly. Tempers would be frayed. Isolated violence would break out in back-streets.
The more canny planners would attempt to create a fire break, pulling down buildings in the direct path of the fire, in order to save those further ahead.
However, all of this would be leading to a huge problem. It was November and therefore winter. The days and nights would be very cold in England, hence not great for suddenly finding yourself homeless. In 1666, Londoners camped upon the slopes of Primrose Hill. This would not be an option in 1605, unless they all wanted to freeze to death.
The first major problem after the injuries and fire would be shelter. Those who could afford it would leave for homes outside London - their own residences or staying with friends. Hotels and inns for miles around would enjoy a profiteering boom. The ugly side of human nature possibly might see prices rising steeply for over-crowded rooms.
My guess is that the majority of people would cross the River Thames. Remember that this isn't an ordinary fire. Westminster Palace has just been exploded and no-one would yet know why, or what was going to happen next. But north of the river would seem much more dangerous than south of it.
The heroes of the hour might well be in Southwark. This was an area filled with theaters, brothels and other houses of ill repute. Shakespeare's Globe Theatre was just down the road, and he would be an eye-witness. Serious civic brownie points and possible goodwill in the future might come from opening up rooms and providing beds there.
Churches would also be great centers for asylum, which would be perfect for all concerned in the circumstances. It would place the displaced in a position where they could hear news; and calm those who thought that this actually was the end of the world.
However, there were no fireplaces nor other sources of heating in those large stone buildings. Those huddled in the pews would be shivering come night-fall.
Eventually it can be expected that official refuges would be provided, perhaps White Hall Palace, if it had survived the fire. Law, order and governance in the capital city could not be done while everyone was displaced, desperate and panicking. White Hall Palace (today's Whitehall) was used as a refuge for commoners during the Great Fire of London a generation later.
It would all be done in the name of the King, though whether James or his four year old son Charles, would depend upon if it was announced that the former was dead.