I haven't done the research myself, but multiple historical bloggers, who may of course all be cribbing from one another, variously describe how under the Church of England -- annulments were handled by the ecclesiastical consistory courts, as opposed to divorces, which went through Parliament -- couples had to *prove* impotence or genital deformity at the hands of a surgeon (or, in earlier times, court-appointed courtesans) in order to obtain an annulment on the grounds of non-consummation. Wilfully failing to copulate wasn't grounds to annul a marriage; the presumption of the Church was that any couple who physically could eventually would. Although sufficient political pressure could eventually bring about the desired outcome, as in the case of the Earl of Essex, whose wife obtained a messy annulment on the grounds of alleged impotence in 1612: https://thehistoryofparliament.wordpress.com/2021/04/15/three-degrees-of-separation-alternatives-to-divorce-in-early-modern-england/
All the articles concur that, while it is a beloved trope of historical novelists, the couple who enter into a 'trial marriage' and plan to have it annulled on the grounds that they never actually slept together simply wouldn't able to use this as a painless, scandal-free method of separation.

• The idea that a couple could get their marriage annulled if they didn’t consummate it did not apply in Regency England, nor to my knowledge, at any time in England.
  
• not consummating the marriage was not grounds for an annulment in this historical period
  
• back in Georgian and Regency England, the fact that the marriage had not been consummated, if it could be proven, was not grounds for either divorce nor annulment.
  
• I know you have seen it in numerous romance novels, but non-consummation was not grounds for an annulment.
  
• The problem is that in that day, non-consummation was not grounds for an annulment but inability to consummate– impotence- was. The man had to be examined and prove that he was incapable of consummating the marriage.
  
• Though a convenient plot device for romance writers, non-consummation of marriage was not considered a reason for annulment. Only the possibility of consummation was required.
  
• A valid marriage could not be annulled just because it had not been consummated. Impotency was grounds for annulment, but this was rarely claimed.
  
• consummation of the marriage was not necessary in England
  
• One of the most insidious faulty facts is the possibility of annulment because the marriage hasn't been consummated yet, that is, the couple hasn't had intercourse after their nuptials. This is simply not true.
  
• A marriage could not be annulled on grounds of non-consummation. (Unless it was as a result of some functional incapacity)
  
  

There may also be differences depending on which church someone belonged to. eg. Catholics and Protestants may have had different rules from each other in some periods.
The Church of England, thanks to its origins, was basically far more Catholic in its attitude than other Protestant churches (it remained the case into the 20th century that a divorced man could not remarry in church, since the sacrament of marriage was considered theologically indissoluble).