The presence of the vaccine remains effective against the new variant because of the broad immune response it elicits.  That is, changes or mutations in the virus are unlikely to render the vaccine completely ineffective.
Countries with high vaccination realization experienced a decrease in patients with severe symptoms, the number of hospitalizations, and deaths.  This is the case in the UK, and the US.  Based on data from the Financial Times, the three countries have vaccinated around 50% of their target population. 
So vaccination does not cause a spike in Covid-19 case deaths.  In some countries where vaccination programs are running well, vaccines actually reduce the number of patients with severe symptoms, the number of hospitalizations, and deaths.