Qatar passenger ops I think will be a matter of when rather than if. Emirates had already achieved what they could do with slots maxed out on A380's at LHR and LGW, so STN was the next logical step. From my understanding, I don't have the full marketing insight of course, but the more popular connections from DXB using the Emirates STN route are places like Thailand, Bali, the Maldives and Mauritius, though Hong Kong and India are quite popular ones too. There is a good leisure and premium demand from north-eastern parts of London that has strengthened over the years, and of course the wider East of England region that the Emirates STN route draws in it's own right, rather than simply an overspill to LHR and LGW.
Qatar on the other hand have managed to get their hands on quite a few additional slot pairs at LHR, and given their strategic importance within the One World alliance, not to mention now matching Emirates on frequency at LHR, that understandably has been their priority. My thinking is either an additional LGW or a new STN service will come once they retire their remaining A380 fleet. The 777-9's that will eventually replace them will no doubt have a high First/Business config on LHR but will have fewer economy seats than the A380s or some of the other aircraft they may replace. That I think will drive the decision making on a potential LGW increase/STN service to replace that capacity, especially around the hub banks in DOH. I think a third daily, evening service at LGW will come again before they look at STN, but it again depends on slots and hub connectivity through DOH.