Well the answer is "No but Yes". The Loyalties can't change and do not change over night. At the same time, when two large powers are fighting for the crown, minor Lordlings may side with one or the other to achieve their own goals. Some prudent ones side with all sides to ensure that their house survives the struggle regardless to the outcome.
I'll divide my answer into following parts to underline the main points here and will link them up in conclusion to provide more causal readers with a summary:
What is the difference between Crownlands and other Administrative units?
You are correct that Crownlands are royal fief. This essentially mean:
- Crownlander Lords fall under direct command of the King.
- There are no central figures among them who can unite them into a formidable force other than the King himself.
This makes the position of the Crownlander Lordlings very much different than rest of the Kingdoms. Other Kingdoms can and have defied or forsworn allegiance to the King in the past because they all have their own Lords Paramount who can rally them into one force. The contact of those Lords with the ultimate authority, the King, is indirect and flows through their own liege Lords who can choose the sides and factions for them and give them little to no choice.
Similarly, Crownlander Lords have little to no choice except following the King. Given their proximity to KL, the retribution for any defiance is guaranteed to be swift and dire.
They can't challenge the Royal House without support of some other Great House.
Does that mean that Crownlander Lords are Semper Fidelis?
They can and have switched sides as well.
Most notable conflicts in which they have changed sides from one warring party to another are:
- Dance of the Dragons
- The Blackfyre Rebellion
In Dance of the Dragons, Crowlander Lords such as Lord Brune of Brownhallow, Lord Brune of Dyre Den, Lord Crabb and Lord Darklyn chose to side with Rhaenyra Targaryen instead of the King in Red Keep, Aegon II. They were not alone in such defections either. For example, Lord Rowan and Lord Beesbury stood for Rhaenyra even though their liege lord Lord Tyrell was fighting for Aegon II.
In First Blackfyre Rebellion, we do not know much about the standings of most Houses. However, we do know that it split the realm effectively into two so there must have been many Crownlander Lords supporting Daemon instead of Daeron or both.
And of course, the famous Defiance of Duskendale, in which Lord Darklyn refused to pay taxes and had the King imprisoned by treachery.
Does Loyalty change over night from Dragon to the Stag?
No it does not.
Despite Robert's almost good rule, there were many who despised him. Robert knew it himself as well.
The king shifted uncomfortably in his saddle. "Perhaps. There are
ships to be had in the Free Cities, though. I tell you, Ned, I do not
like this marriage. There are still those in the Seven Kingdoms who
call me Usurper. Do you forget how many houses fought for Targaryen in
the war? They bide their time for now, but give them half a chance,
they will murder me in my bed, and my sons with me. If the beggar
king crosses with a Dothraki horde at his back, the traitors will join
AGOT - Eddard II
Targaryen still have many supporters who will rise if a Targaryen pretender could land with a strong enough force to take on the Royal Armies. It is plausible that many of such loyalists will be present in the Crownlander Lords as well, just as there are in every corner of the Seven Kingdoms.
Some were even bold enough to show their coldness to Robert:
The royal party had made themselves the uninvited guests of its lord, Ser Raymun Darry, while the hunt for Arya and the butcher’s
boy was conducted on both sides of the river. They were not welcome
visitors. Ser Raymun lived under the king’s peace, but his family
had fought beneath Rhaegar’s dragon banners at the Trident, and his
three older brothers had died there, a truth neither Robert nor Ser
Raymun had forgotten. With king’s men, Darry men, Lannister men, and
Stark men all crammed into a castle far too small for them, tensions
burned hot and heavy.
AGOT - Eddard II
This states that the behavior of Darrys was not friendly to the Royal Escort due to open dislike of Robert's reign and the fact that there was blood feud between the two. Then again, Darry's aren't crownlander but instead they are Riverlander who had defected from the side of their Liege Lord Tully in Robert's Rebellion.
We do not have much information in this regard about Crownlander Houses as they are very minor and play little to no role in the story's progression. We won't see their true feelings until Daenerys lands, if she ever leaves Meereen, that is.
Will they feel the same about Lions taking over?
They don't have to feel anything. Officially, the King is still a Baratheon. They surely must have heard rumors and received letters from Stannis declaring that the King was a lion in a Stag's skin but that doesn't prove anything. But if eventually, the Lannisters do take over openly, Crownlanders will have no choice to grudgingly accept their rule just as they did for the Baratheons. Unless a power large enough to challenge them takes the field and has a viable chance of winning.
- Crownlanders are not very relevant or powerful on the chesstable of Westerosi politics. They do not have a central figure other than the King who can unite them into a Formidable force. So generally they have to accept everything coming from their Liege Lord, which is the King himself.
- Crownlanders have switched sides in the past from the King to the rebels.
- There are still many people in seven Kingdoms, specially in Crownlands, Reach and Dorne who detest Robert and his reign. Dorne even tried to rise up for Viserys as soon as Robert took KL.
- Loyalties indeed die hard.
So in short, Crownlander lords are first and foremost loyal to themselves like anyone else. Then there are other factors such as fear of retribution due to proximity to KL, Honor, Lack of an alternative leader to unite them against the King etc which generally keeps them loyal to whoever holds KL but that's not definite. They have and can change sides to whatever faction suits them.