If the PoD is in the 100 years war, with an English Win, (ideally a total victory in 1351 - or a sneaky treaty with the nobles to institute the system below instead of paying the random) being "British" will inevitably be different, but if we want it as close to British as possible, whilst maximising the size of the future Empire, I'd suggest we have a King/advisor who causes the implementation of Viceroy system over France. Non-inheritable titles that are granted to a Lord, who basically administers as a Duke of Dukes on the Kings behalf. Viceroy of France (Northern France, granted first to Normandy), Viceroy of Aquitainia (Given to either a family member in Aquitaine, or a Gascon lord), and Viceroy of Arles/Bay of Lions (South Eastern and Mediterranean France). This isn't a replacement of the Feudal system, but a way to manage its application. Add some incentives to ensure good decentralised governance, and that system should up.
The benefits of this is that it still allows the manpower of France, with a safer British capital in London - with the addition of the interests of Britain still needing to be attained (Quick, invade the Scots and Irish with Ango-French troops!). It also means that Britain is involved in the Med, at least tangentally, and would be a contender for explorers interested in going out West. (Plus potentially involved in Italian adventures). It also means that (butterflies permitting) Britain is involved in the Med before the fall of Constantinople - which means they could learn/import some Byzantine scholars into S.France and onwards to London - creating the window of a Byzantophile King that starts to transform London into a "Third Rome". (I choose that idea on purpose).
Early discovery of the new world, the beginnings of widespread sewerage and intention towards empire could lead to vast early colonisation in OTL US Eastern Seaboard, and the Caribbean, taking it before the Spanish if the British get the knowledge first - perhaps sharing it with the Portugese in an alt Treaty of Tordesillas, with Spain limited by the British Empire being able to threaten them at land and sea by being France and Britain.
Once the New World starts getting secured, assuming Britain can maintain control in France and the Isles, whilst expanding overseas, it is a powerhouse that can really get the ball rolling if it intended to. A Britain with a strong French influence and Byzantine/Roman undertones could be an interesting cultural phenomenon that has the manpower, wealth and position to take OTL British Empire, and pat it on the head condescendingly. (Mainly because, lets be honest - would this Britain not take India? Doubtful.).
Anglo-Gascon-French in place of English, and Greek/Latin in place of French for posh/languages of the elite? That would be a very different British Empire. But one I'd like to visit.