This is a pretty interesting question with a ton of different dimensions (social, political, economic) to it. Standard diclaimer applies: Medieval western Europe is a big time period, hard to make general statements that are accurate to every locality, etc.

Now, I haven't studied the Wars of Roses in much depth myself, but there wasn't a hugely substantial difference in how English armies were raised in the Wars of the Roses vs the last stages of the Hundred Years War. For one, pretty much all troops would have been recruited and serving under some kind of contract. Contracts of indenture specified terms of service (including pay, compensation for horses lost, division of spoils, how ransoms were handled, etc.) Retinue soldiers would be the personal troops of some aristocrat (including the king). A small percentage of these men would be the aristocrat's personal household men, either serving in his permanent garrisons or simply accompanying him in his daily business. When called to war, they would be some of the best equipped troops available (whether cavalry or infantry).

The rest of the army would be soldiers who entered contracts specifically arranged for a single campaign or term of service, which could be anywhere from a few months to a year or so. Quite a few of these contract troops were experienced professionals who had served on multiple campaigns. These men were contracted through networks of professional captains who could draw on men who had served with them before and also locate/impress aspiring/unlucky new recruits. Because so many of these men were professionals or at least combat experienced, this was a pretty quick means to assemble an army, if the king/aristocrat had the cash to entice men to service. Once word got out that a campaign was being prepared, out-of-work archers and men-at-arms might form companies and head to assembly points where they could enter into contracts and get assigned to retinues. Even individuals or very small groups could get themselves a contract without needing to belong to a larger unit.

Mercenaries and foreign allies were also effective means of raising troops, although both had their issues. Foreign allies were susceptible to changing political circumstances: an allied duke might die and be replaced with an heir less interested in assisting with someone else's battles. However, some political relationships proved to be very reliable. Hundreds of Scottish soldiers died in the service of the kings of France. The line between "foreign ally" and "mercenary" could be complicated. The precise meaning of "mercenary" in the middle ages is complicated in itself, but I won't get into that now. Mercenaries were, contrary to stereotypes spread by writers like Machiavelli, often very reliable and useful. However, it was hard to raise enough mercenaries to form a complete army that could campaign. More often, mercenaries were used to supplement a larger force or provide specialized skills (siege engineers, crossbowmen, gunners).

Now, the options you mentioned above (king raising his own troops vs rallying the nobility) were not mutually exclusive. Any leader embarking on a campaign would attempt to do both. If the political will to fight could not be assembled, it would be difficult to mount a full campaign. A popular and politically skilled king, like Henry V, could marshal Parliament behind him, convince his nobles to prepare their own retinues, and convince the mercantile class/civic authorities to lend him money even if the royal finances were lackluster. Of course, this is all with the intention of fighting a major campaign like Henry V's 1415 invasion of France. A smaller action, like leading a minor raid or repelling small parties of marauding Scots, could be handled with smaller forces. Plenty of minor fights happened (especially along border regions) with the only participants being the local city's garrison or a local lord's household troops.