How did the Tribes Britain transform into the feudal system? And did it happen over centuries?

Anglo-Saxon chiefdoms and early kingdoms didn't as much transformed themselves into feudal states, than eventually, after a period of territorial constitution and political reinforcement, knew a parallel evolution to what happened in the mainland which was cut short by the Norman conquest which imposed sort of an idealised feudality upon the region (altough with various influences from late Anglo-Saxon England).

First, we should agree on a definition of feudality, would it be for the sake of clarity : the word and concept are often used without much consideration for anything remotely medieval, but it end up being irrelevant this way. I think it's best if we consider feudality as the fusion of regional political power and effective landed property. If we go by this definition, feudality began to appear in the mainland "only" with Peppinids/Carolingians hegemony.

While early Anglo-Saxon chiefdoms were roughly based off Late Roman and Britto-Roman ensembles limits, their tribal identity arose from communities whom origins could be found from all the North Sea, Norway to Gaul, rather than being really "imported kingships" which incidentally forced Anglo-Saxons to rely on a mythified past.

By the VIth century paramount leaders and families emerged, and took the ascendency on small regions (some being named according a person, such as Hrontingas, Hrota's people). These were really limited ensembles, 5 to 10km in length, but it was a basis for dynastic power to emerge, with a first set of obligations and tributes from farming communities, with central households appearing over a quite dispersed population that was still tied over familial and productive links (such as transhumance).
Eventually, some families, trough sheer pressure, warfare or else, took control over their neighbours : while it was hard to takeover whole communities, you just had to take down the family in charge to gain dominance, leading to regional territorialisation, concentration and mobilisation of both power and wealth. By the late VIth claims to kingship and regional hegemony, first in South Anglo-Saxon Britain (Kent, East-Anglia, Sussex)
Why there? Because this region played a main trade role in Britain, connected with North Sea and Channel trade with Frisians, Scandinavians and Franks, thus dominating the trade of prestige goods from these places and beyond (dresses, jewelry, swords, tableware, etc.) and their redistribution among Anglo-Saxons dominating families. This trade and redistribution certainly played a great role into making the first Anglo-Saxons kingdoms worth of the name.

Relation with Merovingians probably was a factor too, altough it's not a clear one (Merovingians claimed overlordship of Anglo-Saxons, which was accounted for Papacy and Byzantines, but at the possible exception of Kent, evidence is lacking) : consumption of Frankish or Frankish -carried goods alone isn't much deductive, but the use of Merovingian coinage in Kent, Sussex and East-Anglia hoards does point to a certain infuence in the region, especially in Kent.
The union of Aethelbert and Bertha, daughter of Caribert, probably solidified the royal legitimacy of its dynasty; as well Christianisation of their kingdom : Franks, having inherited the essential of Roman administration, power and prestige in the region, represented a political pole for these early kingdoms which, altough far from blindly and elaborating their own features, borrowed features from Frankish Gaul such as taking inspiration from Salic Law.

By the late VIth, early VIIth century then, we can consider Anglo-Saxon England to be made of dynastic kingdoms, with attempt at royal hegemony backed with capacities of mobilisation, organisation of trade, and Christianisation (there was a Frankish influence there, but less important than inner dynamics based on Irish and Papal missions from one hand, and from remaining British familial influence)
Kings of Sussex, Kent and East-Anglia managed for short period of time to see their authority acknowledged in important part of Britain and while this cyclical high-kingship collapsed regularily, it created structural frames for the next attempts to be more successful. Eventually, kings outside the first region of hegemonic power could take the lead, especially Northumbria and Mercia which beneficed from the same assets while also managing to increase their capacities of political mobilisation thanks to the gains obtained from the conquest of British kingdoms (for instance, Northumbrian kings conquest of the Old North, which led to a nobiliar Anglo-Saxon takeover of these regions).
Basically, powerful families got more powerful, and began to exerce their power into other royal families, without really managing to hold on it definitely as it was essentially based on a personal relationship, to be dealt with at each change of person or even situation.

In the late VIIth century, Anglo-Saxon royal authority met sort of a threshold as warfare still characterized a good part of it, but became less economically interesting due to the redistributive and mobilising needs.
Royal power itself grew from a familial predominance to a broad political power over a kingdom : kings, as it was the case on the mainland, were arbitles and decision-makers. Not only in strategical and redistributive matters, but as well religiously as pointed by the Synod of Whitby, on law-making, town and emporiae build-up and regulation, etc.
Productive benefices still came mostly from tributes and gifts, albeit on a larger scale, which allowed significant hoarded surplus to be exchanged, monetarized, tolled, etc. making trade a necessary compoment of political organisation, with still a significant influence from mainland there (silver minting, for instance) as in other matters such as clerical and scholarship (Anglo-Saxons clerks playing a main role into Carolingian missionary policies) which bounced back in England as a remote imperial model, but overall, it was rathe the epithome of the previous centuries on a larger scale altough some innovations began to appear such as landholders owning services, re-minting of foreign cinage (altough it might be in accordance with Carolingian practice).