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You can get buses via Kampala and Kigale but it's a long journey (24 hours to Kigali and then another 6 to Bujumbura). The buses won't cost much (maybe 30GBP in total) but you'll also have to pay for a visa for Uganda (50USD). The last time I took this journey, in 2011, the bus killed a woman on the road. Travelling through Uganda by bus is not for the faint-hearted.

There are several buses a day between Nairobi and Kampala. The journey takes about 12 hours, I think. They tend to leave early in the morning or late a night. Some go all the way to Kigali (another 12 hours). From Kigali there are many departures in minibuses (run by companies such as "YAHOO!" (not the real one)) to Bujumbura.

I know it seems expensive but, in an African context, a flight for 200GBP is not horrendously so. There's no Easyjet or Ryanair in this part of the world.

If you haven't already booked your flight to Nairobi, you could try to search for flights on "multicity" sections of websites like Opodo and Expedia and put in the entire itinerary, including the Burundi part. Sometimes you can save a lot of money by including this type of side trip in your main ticket. So your journey might be LON-NBO-BJM-LON, for example.