The only direct flights seem to be with Air Zimbabwe, from Gatwick to Harare. Their website is offering return flights on that route 'from £300, excluding taxes' but I keep getting error messages when I try to search for specific flights and prices:
http://www.airzimbabwe.aero/
This site claims to be an 'authorised dealer' for Air Zimbabwe. They're offering (for some random June/July dates which I chose) a return fare of £439 plus taxes. (I can't check on the taxes because the site doesn't offer online booking. You have to phone up):
http://www.airzimbabweuk.com/
Flights with other airlines require connection. BA, for example, will take you via Jo'burg, with a much higher fare. (The cheapest I've found is £927.23 but that does include all taxes and charges)
From the BA site, these would appear to be the charges which you might need to add to the Air Zimbabwe fare:
UK Air Passenger Duty £75.00
UK Passenger Service Charge £30.63
Zimbabwe Passenger Service Charge £21.60
(I would hope that Air Zimbabwe don't try to hide any fuel surcharges, or other airline charges, under 'Taxes')
Chris