Put simply, Welsh culture didn't survive that well at all. To start, the Welsh experience is probably closest related to the experience of the Cornish in regards to both having been directly a part of England and English legal structure for an extended duration of time; Wales was initially caught in a cycle of on and off English control (An English monarch would attempt to conquer Wales, the Welsh would submit but immediately rebel the moment they saw the chance, repeat), a process that would continue with varying success until the 1280s when Wales would finally be fully annexed into England. Initially, Wales was left with its own legal framework and was largely ruled by the same lords as previously, something that would gradually and then completely change when the Laws in Wales were passed in the mid 1500s, when Wales was directly integrated into the English legal framework, essentially eliminating the Welsh state. This is distinctly different from the Scottish experience, with Lowland Scots already largely having spoken languages of Anglo-Saxon origin even before the Union of the Crowns, be it Scots or Scottish English, and with Scottish Gaelic being largely confined to the highlands and gradually wiped out through the period of the Clearances (Scots would also begin to die out during this time), or the Irish experience which can be far closer amounted to colonialism and cultural genocide, and is far too large of a subject to tackle. To get back on initial topic, Wales as an idea and as a people would gradually fade out of memory over the next centuries; Welsh Celtic was replaced by English nearly everywhere except for extremely rural places in the northwest, and the south especially became indistinguishable from the rest of Southwestern England. This wasn't necessarily performed out of malice; English culture was so dominant that other measures weren't really necessary and the elimination of the Welsh state was enough to initiate this process. However, this began to change towards the start of the Industrial Revolution, and the influencing factors brought with it. Before the industrialization of Britain, Wales was a quite poor agricultural region, but Wales contains a massive amount of both iron and coal, and as Wales industrialized, both her population and local wealth exploded, and local wealth brought influence over her own politics. The first laws explicitly exclusive to Wales would begin to appear in the late 1800s, and Welsh culture began to rebound on the backs of this newfound influence Wales held in Britain. However, this was a gradual process; the Welsh voted overwhelmingly against self rule when first asked in 1979, but would barely vote in favor in 1997 (Compared to the Scottish, who voted overwhelmingly in favor of this). Since Devolution, Welsh cultural resurgence has continued, especially in the regions where Welsh culture was previously still existing, and this is largely due to the desires of the Welsh themselves.

I'd argue Breton culture actually survived longer than Welsh culture in its natural state; Under the French Monarchy, there were no attempts to change French minority cultures and unless you were a bureaucrat or noble, you weren't expected to speak French. However, the French state made massive efforts in the middle-late 1800s to eliminate regional dialects and cultural identities to centralize French culture around the Parisian standard, which is largely when Breton language and culture would fall out of favor. Another interesting topic, but also a different subject entirely.