Was there a near-constant train of men moving toward Jerusalem?

No, not at all. While people did arrive from Europe to the Holy Lands (particularly after the Crusader States had already established themselves) separate from the main Crusader armies, for the most part the Crusader Armies, as they gathered in Constantinople, were pretty much set for the rest of the Crusades. Of course, huge amounts of them died or deserted before ever reaching Constantinople, and more of them would die throughout the rest of the Crusade. Here is a map of the Crusade starting with the formation of its armies in Europe and ending with the capture of Jerusalem (or technically the Battle of Ascalon a little bit after that). It was a rather bloody affair, and the force that ended up taking Jerusalem was a hollow shell of what originally departed from the various locales around Europe (most of the Crusaders in the First Crusade were from France, Southern Italy, and some parts of Germany).

The Byzantine Army also played a role in the First Crusade, particularly in negotiating the capture of Nicaea in 1097. Byzantine soldiers under the general Tatikios accompanied the Crusaders all the way to Antioch, but Tatikios left the siege of that city in early 1098 to bring reinforcements (he never did this, as on the way back down through Anatolia, the Byzantines were erroneously informed by high-ranking Crusader noble, Stephen of Blois, that the Crusaders had been wiped out by the Turks). Byzantine supplies were vital to the Crusaders early on, though they reduced after Bohemond crowned himself Prince of Antioch rather than returning it to the Byzantine Emperor as the Crusaders had sworn to do (Bohemond justified this by claiming that the Emperor broke his oath to help the Crusaders when he didn't send reinforcements to Antioch).

The lack of manpower was a huge issue for the Crusader States as they tried to establish themselves. Now, I should note that some bits of the Crusader army split off on the way to Jerusalem to settle other areas. The Crusader states known as the County of Edessa - established by Baldwin I, the younger brother of Godfrey of Bouillon, who he would succeed as leader of the Kingdom of Jerusalem - and the Principality of Antioch - established by the charismatic Norman Bohemond of Taranto, much to the consternation of the Byzantine Empire - were both created in 1098 Baldwin had separated from the rest of the Crusader force to take Edessa, whilst Bohemond managed to get the other Crusaders to give him Antioch after they captured following a long siege. If you had attached yourselves to Baldwin or Bohemond, you may very well have not actually journeyed to Jerusalem itself until the city was captured by the remaining Crusader army in 1099.

The Council of Clermont which called for the Crusade was in November 1095, and most of the Crusaders set off the following year. If you had set off when the Pope had planned the departure to be (the Feast of the Assumption in August 1096) it would have been near 3 years before you made it to Jerusalem - providing that you did make it to Jerusalem. Now, you asked about "England to Jerusalem," but the thing is, the First Crusades really didn't have many English participants in it. I can't think of any. The Duke of Normandy Robert Curthose, son of William the Conqueror, partook in the Crusade, but he left from, well, Normandy.

What sorts of non-combatants followed the soldiers?

All sorts of individuals. The Crusade was a pilgrimage to the Holy Land that also happened to be rather heavily armed. Besides the knights and their soldiers, though, was an array of all sorts of other people. Members of the clergy high and low, hermits, peasants (many of whom might have styled themselves as soldiers despite possessing no arms or skill of any significance), craftsmen, et cetera. Thousands of women were a part of the Crusades, and even a substantial amount of children. In hindsight, this wasn't exactly the best idea (most of them ended up dead or enslaved), but you have to remember that the First Crusade was a highly religiously motivated campaign, and knights and lords held no monopoly over religious zeal.

In the Byzantine-controlled Balkans, the Byzantines had arranged for guides (mainly Pecheneg mercenaries) to escort the Crusaders. The Crusaders had a bad habit of attacking the Pechenegs, though.

While most of the Crusader armies were lead by important men like Raymond of Toulouse, Godfrey of Bouillon and his brother Baldwin, Bohemond of Taranto, Robert of Flanders, and Robert Curthose (all of whom had extensive holdings in Europe, with the exception of Baldwin and Bohemond on the virtue of them having older brothers), there was a separate army that heeded the call for the Crusade, mainly led by a charismatic priest named Peter the Hermit and called the People's Crusade. Now, despite the name, there were lords and knights who followed Peter the Hermit as well. In general, the People's Crusade was a pretty good cross-section of European society at the time. The People's Crusade, however, probably did more harm to the Crusader cause than good. Along with the forces of a German Count named Emecho they massacred Jews in the Rhineland, plundered towns and lands in Hungary which made them an enemy of the King of Hungary (who ended up defeating killing quite a few of them), and when they eventually arrived in Constantinople before the other Crusader armies, they insisted upon being allowed to enter Asia Minor. The Byzantines shipped them across to Asia Minor and most of them were killed or enslaved by the Turks. Probably their only real positive contribution to the Crusades was that this might have lulled the Turks into a false sense of complacency, since the armies that followed them were much, much better led and equipped.

It would normally take pilgrims a few months to reach Jerusalem from western Europe (depending on where you were and the obstacles you faced along the extremely perilous journey), but the first crusaders were not ordinary Pilgrims (despite referring to themselves in several documents as such). The sheer size of the army, and the accompanying scale of battles, foraging, weather, disease, and all other manners of problems slowed them down immensely. Interestingly, none of the primary records focus on it taking an unexpected amount of time, which may suggest that they were prepared for a much longer journey than previous pilgrimages. Having been preached in late November of 1095, the crusade took until August of 1096 to launch - obviously with a couple of false starts along the way, like the ill-fated "People's Crusade" (the proper crusade is sometimes called "The Princes Crusade"). It was delayed in reaching its destination of Jerusalem (which did not become the goal of the crusade until some time along the way, months after its original preaching) with a siege of well over a year on the critically important city of Antioch. By the time the crusaders stepped foot over the walls of Jerusalem in July of 1099, they had been on their journey for three years.

As for how they got there, they walked (or rode) until they got the Byzantium, they boats across the Bosporus, then more walking until they got there.

There was always a near constant train of pilgrims after this, but they really took off in 1101.

As for your second question, there were thousands of non-combatants who came along with the crusaders. While most crusaders came alone, many brought some or all of their families with them, too. Cooks, entertainers, thieves, prostitutes, beggars, merchants, and blacksmiths can all be assumed to have accompanied the crusaders for portions of the journey, many all the way. Most importantly of all, the crusade was preached, thought of, and externally portrayed as first and foremost a pilgrimage - this meant that noncombatants were inspired to take up the pilgrimage as well. These noncombatants were extremely important for the crusade and they were sometimes even involved in the heat of battle. Most notably, the noncombatants suffered dearly at the battle of Dorylaeum (1097) where the crusaders were caught off-guards by a surprise attack, leading to the non-combatants being swarmed and many massacred by Seljuk cavalry and horse archers.

sources:

Fulcher of Chartes

Anonymous author of the Gesta Francorum

Raymond d'Aguilers

Albert of Aachen

Peter Tudobode

(all chroniclers of the first crusade)

I should just like to point out that was little or no actual English support in the First Crusade. No Anglo-Saxons and none of the England based Normans that I've found evidence for. So the march to Jerusalem started on the mainland side of the channel.