Whether it's Warwick or Warwick Parkway, Leamington Spa is less than 10 minutes away by train so just hop onto a train going there (if you're southbound presumably it'll be the stop before Warwick). Then get a pink Stagecoach U1 bus that goes up the Parade and to campus, takes 25-40 minutes depending on traffic.
Or if you're going southbound and you pass through Coventry before Leamington, get off there. Travel Coventry buses (you want the number 12) stop on the bridge overlooking the station so get on there, they take about 20-25 minutes generally.