As Snailplane noted in a comment, the correct word in this case is increase:
Old response time: X time units
New response time: X+Y time units
The response time is greater than it was before the change, so it has increased:
After implementing the new system, response time has increased.
However, this can be ambiguous. Not because of the meanings of the English words, but because of how readers interpret them. In a number of contexts, an increase is positive:
Our earnings have increased 5% over last quarter
Hard drive capacity has increased greatly over the last ten years.
It is possible that readers will see the word increase and ignore context, instead just assuming that it's a positive. To avoid this, use a word like worsen or hurt, as Jim noted in another comment. These words are unquestionably negative without any ambiguity.