The physical installation of a radiant infrared system in a typical parish church takes between two and five days. Wall or ceiling-mounted heaters need fixing points, cabling back to a distribution board, and connection to a control system. It is not a major building project.
The longer timeline is everything that happens before the installers arrive. Getting a faculty can take three to six months depending on your diocese, and if the DAC requests additional information or a site visit, add more time. A straightforward List B minor works proposal for heating can sometimes be handled under delegated authority, which is quicker, but most new heating installations need a full faculty.
The practical advice is to start the paperwork early. Get your survey and quote, talk to your DAC secretary informally, and submit the faculty petition as soon as you can. The installation itself will probably be the quickest part of the whole process.