How long did it take for American and European churches to accept the theory of evolution?

The first official papal statement on evolution came in 1950 from Pius XII in Humani Generis. Most people will quote sections 36 and 37 and uncritically take them purely at what they say, but this robs the statements of their context. For instance, section 8:

In all this confusion of opinion it is some consolation to Us to see former adherents of rationalism today frequently desiring to return to the fountain of divinely communicated truth, and to acknowledge and profess the word of God as contained in Sacred Scripture as the foundation of religious teaching. But at the same time it is a matter of regret that not a few of these, the more firmly they accept the word of God, so much the more do they diminish the value of human reason, and the more they exalt the authority of God the Revealer, the more severely do they spurn the teaching office of the Church, which has been instituted by Christ, Our Lord, to preserve and interpret divine revelation. This attitude is not only plainly at variance with Holy Scripture, but is shown to be false by experience also. For often those who disagree with the true Church complain openly of their disagreement in matters of dogma and thus unwillingly bear witness to the necessity of a living Teaching Authority

This encyclical is about faith and reason, as well as the limitations of those two things. Pius XII is making a series of theological and philosophical statements in this document--not dictating science. There are thirty-four sections of argument before we even get to the portion about evolution, which is very much not the featured portion of the document (which is much more about Nouvelle Théologie than science and its implications on theology).

Anyway, here is sections 35-37

(35) It remains for Us now to speak about those questions which, although they pertain to the positive sciences, are nevertheless more or less connected with the truths of the Christian faith. In fact, not a few insistently demand that the Catholic religion take these sciences into account as much as possible. This certainly would be praiseworthy in the case of clearly proved facts; but caution must be used when there is rather question of hypotheses, having some sort of scientific foundation, in which the doctrine contained in Sacred Scripture or in Tradition is involved. If such conjectural opinions are directly or indirectly opposed to the doctrine revealed by God, then the demand that they be recognized can in no way be admitted.

(36) For these reasons the Teaching Authority of the Church does not forbid that, in conformity with the present state of human sciences and sacred theology, research and discussions, on the part of men experienced in both fields, take place with regard to the doctrine of evolution, in as far as it inquires into the origin of the human body as coming from pre-existent and living matter - for the Catholic faith obliges us to hold that souls are immediately created by God. However, this must be done in such a way that the reasons for both opinions, that is, those favorable and those unfavorable to evolution, be weighed and judged with the necessary seriousness, moderation and measure, and provided that all are prepared to submit to the judgment of the Church, to whom Christ has given the mission of interpreting authentically the Sacred Scriptures and of defending the dogmas of faith.[11] Some however, rashly transgress this liberty of discussion, when they act as if the origin of the human body from pre-existing and living matter were already completely certain and proved by the facts which have been discovered up to now and by reasoning on those facts, and as if there were nothing in the sources of divine revelation which demands the greatest moderation and caution in this question.

(37) When, however, there is question of another conjectural opinion, namely polygenism, the children of the Church by no means enjoy such liberty. For the faithful cannot embrace that opinion which maintains that either after Adam there existed on this earth true men who did not take their origin through natural generation from him as from the first parent of all, or that Adam represents a certain number of first parents. Now it is in no way apparent how such an opinion can be reconciled with that which the sources of revealed truth and the documents of the Teaching Authority of the Church propose with regard to original sin, which proceeds from a sin actually committed by an individual Adam and which, through generation, is passed on to all and is in everyone as his own

35 urges caution in applying science to theology, and asserts that some theological principles have preeminence in the realm of theology. 36 gives qualified permission to accept evolution, provided that the theological question about souls be left to theology. 37 is bandied about by those attempting to "prove" that Catholicism rejects evolution, but its ban on polygenism is narrow (as are all bans in Catholicism). It is also a theological point, not a scientific claim. Further, while section 36 has been referenced multiple times by subsequent pontiffs and section 35 has been echoed many times as well there has been nothing in the way of pontifical reinforcement of section 37. That doesn't obviate its existence or its relevance, but it indicates that subsequent pontiffs have placed more emphasis on the earlier passages over the latter. Since none of these statements--from Pius XII or following pontiffs--claim papal infallibility they demand the respect of faithful Catholics but do not claim to require their assent (unlike another document from 1950, Munificentissimus Deus which defined the dogma of the Assumption of Mary). As such, a Catholic is free to accept or reject evolution (section 36) and has some freedom to differ regarding section 37's ban on polygenism as a theological assertion.

In short, prior to 1950 the acceptance of evolution among Catholics varied but 1950 is the easiest date to speak of papal acceptance.