It looks like not too many airlines fly to Rangoon for obvious reasons. MH joins Oneworld in December so that may be a bit late. I am surprised that Dragonair doesn't fly there.

Not a lot of Oneworld flights there currently but you could fly BA to BKK or SIN then take another carrier onto Rangoon. I think BA interlines with Thai, although they are in Star so no miles etc..

Jetstar fly SIN-RGN it appears and I believe it has a QF codeshare so you could fly LHR-SIN-RGN on QF codeshare.

A codeshare option is not always important and interlining is fine so long as you leave enough connecting time. Whilst I have zero knowledge about Burmese airlines I wouldn't count them being very safe so I think sticking with respectable carriers like Thai for the 2nd leg is advisable.