I recently gave TOEFL and got 99 with only 20 in writing. However the 6.5 equivalent of IELTS corresponds to minimum 23 in writing. Do I need to redo the test?
Depends on who is asking for it, and their regulations.
Ask them first. Some universities in the UK accepts only IELTS regardless of what else you have at hand. Others look at your overall experience, and ask for an IELTS only if they think your level may be not high enough.
Never heard of anyone asking for TOEFL in the UK, as it is an American test.