It is absolutely true that some abolitionists, both in Europe and the Americas, did all they could to refrain from purchasing goods that were products of slavery, including sugar, tobacco, (eventually) cotton, and others. This was part of a new trend in Atlantic societies that were increasingly consumerist, a "politics of consumption" that has echoed to the current day. In the United States, proslavery Southerners frequently scorned them as hypocrites, and southern magazines and newspapers from the antebellum era are full of stories gleefully reporting that such and such a Yankee abolitionist, "who refuses even to eat West Indies sugar," had been caught, say, participating in the illegal slave trade because of the promise of high profits. Some of these stories were even true.

So, yes, consumer boycotts and changed personal consumer habits were a part of the politics of abolition in Europe and the Americas.

However, the impact of slavery and slave grown sugar on British food culture is in most ways the opposite of what you've described. For one thing, black treacle, and other forms of molasses and molasses based products, are byproducts of sugarcane refining! It would be a poorly informed abolitionist who refused to purchase cones of refined white sugar from the Caribbean but continued to use black treacle in cooking as they enjoyed a nightly tot of molasses-based rum.

For another, your perception that sugarcane products were more popular and purchased in larger amounts by the British public than alternative sweeteners is correct. Caribbean sugar has had a profound impact on British food culture. Tea with sugar became a beloved drink for upper class Britons in the colonial era, and became more widespread and accessible in the 19th century. British baking culture, as celebrated in contemporary tv shows like The Great British Bake Off (which I adore), came from the abundance of slave grown sugar that flooded Europe beginning in the mid-17th century. Some historians have argued that massive consumption of sugar, a high energy food with little nutritional value, helped to fuel Britain's industrial revolution in the nineteenth century even as it led to serious health problems among the British working classes.

Though the population of Britain was more widely committed to the end of slavery than, say, the free population of the United States (for a bunch of reasons), and though Britain actually engaged in what historian Seymour Drescher has called "econocide" by ending slavery in their caribbean possessions when it was at its most profitable, abolitionists who refused to consume products associated with slavery were too small in number to have a major impact on British sugar consumption or food culture.