According to most reports the answer appears to be yes, with both businesses and the public careful to stick to the rules. As expected a small minority abandoned social distancing as the booze began to flow. That resulted in general overcrowding of streets and public spaces in some areas. This may lead to a spike or spikes, which in turn end in local lockdowns. But that is a far cry from the idea that a second wave will flow uniformly over the nation and require a national retreat of the population behind their own closed doors, plus the closing down of much business activity.
Whether that happens depends on test and trace working properly. According to ministers it is functioning well. According to some credible insiders it is not anything like as effective as it should be and needs to be. We must hope the government is right.