Side question: How long would it take a peasant to make the trip if they had the means to?
I'm afraid that we don't have much evidence on the details of medieval travels especially to answer the optional part of OP's question in accordance with different social ranks of travelers during that period (OP might find it interesting to read /u/PartyMoses's and possibly also my answers in I'm a traveler in ad1200 going from Lübeck across the HRE to Rome on secular business; Do I travel alone or in a group? Do we camp every night in the woods or am I in a Tavern/Inn every night?
The closest relevant medieval text I know to the area and period of Alfred's journey is the itinerary of newly consecrated Archbishop Sigeric [of Canterbury] to Rome in 990. On his returning route, the itinerary lists 80 night stops (submansiones) between The City of Rome (urbs Roma) and Sombre by Wissant on the French side of the English Channel (Sumeran). This itinerary is the oldest account of the night stops as well as the route on Via Francigena, a popular medieval pilgrim route extended from Canterbury to Rome (and amounting to ca. 1150 miles in total), and If we mechanically calculate one day for each night stops, it took about 2.5 months to a little less than 3 months.
We should also keep in mind, however, that some possible accidents like bad weather and sickness might also have extended the duration of the travel, and also that the most difficult part of the route, the Alpine pass was closed during Winter. A 12th century pilgrim notes that there was a snow remained on stones and a lake by St. Bernard pass (2,491 meters high) even in the end on July (Kyrkjebø & Spørck (overs.) 2012: 85).
The actual 'business' trip also tended to take a bit longer than the calculation based on the itinerary probably due to the businesses. To give an example, when Cardinal Nicholas Breakspear (later Pope Adrian IV) took a visit to Norway from Italy as a papal legate by way of England in 1152-1154, he left northern Italy in the end of March (witnessing a charter issued in Luni on Mar. 30, 1152) and arrived in Norway probably by the end of August of the same year by boarding a ship from England - in other words, he probably had to finish his business by latest around the middle of August, ca. 4.5 months after his departure in Italy (Cf. Bergquist 2003: 42-44). The medieval journey of the papal legate is rather (relatively) easy to trace due to his attendance to various events like the council and the witness to the document, but for lay person's travel, it became much more difficult to reconstruct the alleged timeline.
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