In the Middle Ages Norfolk became wealthy from the wool trade and one of the most populated areas of Britain. So the Icknield Way, one of several ancient trackways in S England, will undoubtedly have been used by pilgrims from Norfolk south, and from the Thames Valley to shrines in Norfolk, particularly Walsingham.
If you wanted to start in Walsingham, where only ruins remain of the large Augustinian priory and the Franciscan friary (though nearby are the more substantial remains of Binham Priory, a Benedictine cell of St Albans), the former railway line can take you south to Fakenham, where Augustinian Hempton priory used to provide accommodation for pilgrims, though little remains of it. You can reach the next stop, Castle Acre, a Cluniac foundation with some impressive remains, via the abandoned medieval village of Godwick and the Nar Valley Way from Tittleshall. Further west on the Nar Valley Way is West Acre, which had an Augustinian priory. Castle Acre is on Peddar's Way, a Roman road now a National Trail, leading south to Thetford Heath. (Icknield Way was further W.)
Norwich, with its fine Norman cathedral (formerly Benedictine), became one of the largest towns in Britain. See Jonathan Plunkett's excellent page on monastic remains in Norwich; the Dominican friary, now called St Andrew's Hall, is one of the finest surviving in the country. The church of St James Pockthorpe in the N part of the city centre is now used as a puppet theatre.
From Norwich, the next stop for pilgrims would be Wymondham Abbey, where the church of the Benedictine priory (also founded as a cell of St Albans), with its distinctive towers, still serves as the parish church. To the west is Crownthorpe, with an Early English church of St James. The next stop, passing Gt Ellingham where there is also a church of St James (C14), would be Thetford. This town, formerly Danish capital and once the seat of the bishop of E Anglia, has the remains of a Cluniac priory, a further Augustinian priory, and a Benedictine nunnery; there were also Dominican and Austin friaries.
The Icknield Way runs SW from Thetford to Newmarket. Pilgrims though are likely to have diverted S to the shrine of St Edmund at Bury St Edmunds, where the cathedral is now dedicated to St James. The St Edmund Way will take you there.
From Bury head W again to Dalham where you can pick up the Icknield Way Path; much of the actual Way is a major road, so this shadows it on paths and tracks. The chancel of the church at Royston, where the Way crosses Ermine St, is all that remains of the Augustinian priory.
Just before Luton, you have a choice. You can continue on the Icknield Way Path and its continuation, the Ridgeway, along the line of the Chilterns, all the way to the Thames Gap at Goring. This passes through Dunstable, where the Way crosses Watling St, and where the fine Augustinian priory church is now the parish church (see photos here and here) and the priory hospice is now a heritage centre. Parts of the Augustinian nunnery church at Goring also became the parish church. From Goring it is a short walk along the Thames Path to Reading.
An alternative is to use the route to St Albans given in the Ermine St page, and then continue W to Marlow, following the Ver-Colne Valley Walk S to Mundon where it crosses the Hertfordshire Way, following this W to Chipperfield, where the Chiltern Way can be followed again to the Thames at Marlow.
Pilgrims who prefered to go from Bury to London, or more likely Waltham Abbey, rather than Reading could have taken a Roman road which passes E of Bury and via Sudbury to Braintree and Chelmsford. The St Edmund Way shadows this S from Bury via Lavenham to Long Melford and Sudbury. However, little remains of the Benedictine priory or the Dominican friary at Sudbury, so from Long Melford you can turn W on the Stour Valley Path (just to the N, Stanstead has a church of St James). The remains of the former Austin friary at Clare now form part of a community of Augustinian friars who welcome visitors. W of Braintree is Great Saling which has a church of St James the Great. At Little Dunmow with its famous flitch, the church is all that remains of the Augustinian priory, and even less remains of Cistercian Tilty abbey on the other side of Gt Dunmow. Just S of Stansted airport are the scant remains of Augustinian Thremhall Priory, which you can reach from Dunmow via the Flitch Way, a converted railway line. Next, head S along the Three Forests Way to Hatfield Regis or King's Hatfield, now called Hatfield Broad Oak, where the nave of the church is what is left of a Benedictine cell of Rennes. Then head W to pick up the Forest Way which takes you almost to Waltham Abbey.