Let's see what the relevant sections of the Constitution have to say as to how much say the Federal government has over its own elections.
In the case of Congress itself, the Federal government has wide latitude. It could require all States to hold primary elections for Congressional elections on a particular day or days, and it could require the use of some method other than first past the post. Note in particular that the use of the Tuesday after the first Monday in November as Election Day is not mandated in the Constitution. If Congress wished to, they could move election day to the middle of December.
The last phrase pertains to when there were legislative elections for the Senators, and prevented the Congress from telling State legislatures where to meet.
Incidentally, while back in the days of legislative election of the Senate, the legislature could choose to hold a special session for the election outside the State, in the case of the Electors, they can't turn it into a paid holiday in Aruba. This provision avoids the cost and inconvenience of the Electors traveling to the national capital to cast a single vote in person and then going home.
This could be construed as to give Congress authority over primary elections, if such are used, but it would be a very shaky authority. The Constitution does not really cover popular elections to the office of Elector in any detail, and it certainly does not anticipate as convoluted a process as is currently used. In short, while Congress could specify specific Times if a State selected a particular method of choosing its electors, it has no authority to require the States to pick any particular method. And then there is the fact that I see no way Congress could require the political parties to adhere to a schedule.
And then there is the problematic singular "Time". Even if the other difficulties are overcome, I can't see this as being anything other than giving Congress authority to select the date on which Electors are definitively elected, and not any of the preliminaries.
Note that the lengthy period between Election Day and Inauguration is not required by the Constitution. Moving Election Day to mid-December and having the Electors meet on the same day as Congress opens would be perfectly valid and could be easily done if desired.
So we are left with indirect methods of persuasion. Beyond the bribery Bacon King mentioned, there is one other stick that could be used. Congress could use its Article I Section 4 Clause 1 power to require that if States hold a Presidential primary on a particular date, they would have to hold their Congressional primaries on the same date. Most, if not all, of the early voting states have separate dates for their Presidential primaries and all other primaries. Forcing the dates to be congruent would likely see a move to party-funded primaries and caucuses to avoid the congruence.