Pre-Brexit, English had pretty much cemented its status as the lingua franca of Europe. It is one of the three working languages of the EU, along with German and French, but is by far the most spoken in Europe. However, post-Brexit, this could be a different story. Indeed, MEP Danuta Hübner even claimed that English will cease to be an official language of the EU once the UK leaves the Union.
And you can understand why. It would seem strange for countries such as Germany and France, who, without the UK, would hold the two most spoken languages, to support the continuation of English as such a widely-used language within the EU. It would make sense for them to support the rise of French and German. However, it would be no small change. Currently, translation services make up 1% of the EU's annual budget, and with English being understood by 51% of all adults in Europe, it would take years, if not decades, to phase English out. And that's not to mention Ireland and Malta. Although the UK is currently the only country in the EU where English is the official native language, both Ireland and Malta count English as their next official languages, and so by removing English as an official language of the EU, Ireland and Malta would be left in a problematic situation. In fact, after Danuta Hübner's comment in 2016, Ireland released a statement asserting that this decision would have to be put to a unanimous vote, which Ireland would, of course, be a part of.
Since that time, it was confirmed in the European Commission's proposed budget for 2021-27 (in small print) that the English language isn't going anywhere any time soon, stating that English language translation and interpretation services will not be affected by Brexit. Although English may become slightly less used over time in a post-Brexit Europe, the reality is that today English is not so much learnt in order to speak to natives, but rather to communicate with people of different native languages all over the world: a true lingua franca. This means that, realistically, English is unlikely to disappear from the European Union. Moreover, English has risen as a global language mostly though the influence of the United States, its music and Hollywood: English is far bigger than the United Kingdom. As such, it is doubtful that Britain's exit from the EU will significantly change the popularity, or use, of English in Europe or the European Union.