This is a really interesting question, and although I'm not entirely sure if I'm completely equipped to answer this question, I think I can give you a succinct explanation as to what some of the reasons are for language death, particularly in reference to some other, comparable situations. When you say "successful", I assume that you mean "alive" rather than extinct. The question relates to two particular subfields within linguistics: historical linguistics and sociolinguistics.
There is a general notion I need to adress first: the relationship between ethnicity and language is a tricky one, and not as straightforward as it is sometimes made out to be. Especially before the development of nationalism as a philosophy and art form in the 19th century, the language one spoke did not necessarily be an indication of ethnicity (and when you think about it, neither is it now), and the relationship between the two seems to have mostly affected by practicality or prestige. We'll get back to this later as well. It's only at a later moment that language becomes a more important part of identity.
I’m not a historian of the British Isles at all, so maybe someone who knows more about that can tell you more about the historical context in which Celtic-speaking areas were annexed by their respective overlords. In the case of the British Isles, this is by the English, who spoke a Germanic language, and in the case of Britanny by the French, who spoke a Romance language. When we get a map of Europe in front of us, we can see that virtually all of the places where Celtic languages are spoken (with the exception of Ireland, and arguably also the Isle of Man) are governed by countries that have either a single non-Celtic official language, such as French in the case of Britanny; or whose primary official language is non-Celtic, such as English in the case of Wales and Scotland. In short, Celtic-language speaking areas have been under the control of non-Celtic-language speaking powers for several centuries and up until about a millenium.
This is where the two factors I talked about earlier, namely practicality and prestige, come back into play, and historical linguists often refer to the two to talk about loanwords and language death. Firstly, there may be practical reasons to start speaking a new language: usually for economic purposes, such as to trade with other people, or out of or pure necessity, for example, in case of military drafting. Another reason is prestige: the language of the invader is associated with the elite, and learning it can increase chances of upwards social mobility (this, for example, is likely what happened in Egypt and the rest of North Africa after the Islamic conquest), or more negatively put: holding on to your own language diminishes social standing and can lead to persecution. In the case of the Celtic languages, French and English were the prestige languages (known as superstrata).
Considering these circumstances, it might be a lot more interesting to ask why these Celtic languages didn’t go extinct. This can be partially explained through geography: regions where Celtic languages are/were spoken were generally remote or difficult to access, such as the very tip of Britanny or the Outer Hebrides in Scotland, or the western part of county Galway in Ireland. There was generally less contact with foreigners in these areas and hence fewer reasons to to learn different languages, let alone forego the first.
The sociolinguistic element of this question is strongly related to the rise and influence of nationalism, which starts to play a significant role within ethnic communities from the 19th century onwards, particularly in Europe. Basically from the French Revolution onwards, identity become more dependent on ethnicity, rather than religion, which also led to an increase of metalinguistic awareness. In some cases, this was government-sponsored, such as in France, where increasingly intolerant linguistic policy from Paris led to the virtual extinction of many minority languages in France. Some of these languages, such as Lorrain, Gascon or Provençal, could be considered dialects within the same linguistic continuum in France, or at the very least, were typologically closer to Metropolitan French, and these disappeared rather quickly. However, the non-Indoeuropean language Basque and the Celtic language Breton, which are typologically less similar to French, were less strongly affected by this policy. However, some more…passionate French nationalists may still insist that Basque and Breton are dialects of French.
On the other hand, nationalist sentiment in Ireland, Wales, and Brittany has very much served to keep Celtic languages alive in these places. For example, in an 1886 essay, John Fleming, editor of The Gaelic Journal, wrote the following:
“Make yourself acquainted as well as you can with the industrial resources, the history, the language, and literature of the country. You will then be a patriot, and able to serve the country.”
In short, the survival of the Celtic languages in these places can be attributed to a number of factors, including geographical and social ones. I would kindly ask a historian of these areas to provide a more appropriate context for these specific regions.
Bibliography
Campbell, L. Historical Linguistics - An Introduction (Edinburgh:2006), (check out the chapters on models of linguistic change, explaining linguistic change and areal linguistics).
Fleming, J. "The Revival of the Irish Language", in The Irish Fireside, Vol. 6, N. 133 (1866). Accessible online
Wolf, M. An Irish-Speaking Island, particularly the introduction.