This is very specialised information you are asking for, and for quite an extended period of time- if you are going to be working in this area, the best people to ask would be the people you are going to work for. In that area there are a couple of Norman Carr bushcamps (Luwi and Nsolo) and the wild dog research project is active in that area.
I've just spent 3 months working in an area a bit further north than Luwi (round Mwamba/Lion Plains). The only internet access in these areas will be via satellite at the camps and Luwi and Nsolo may not even have that- some bushcamps only communicate by radio with their main camps. If they do it will be restricted to use by the management or guests in an emergency.
Mobile phone coverage- unlikely, and more unlikely the further you get from the Luangwa river. In the area I was in there were a few spots that the local guides knew about where you could just about get a signal on a good day if you stood on top of the vehicle. These spots were mainly close to the Luangwa river, but further inland there was nothing. So along the Luwi river I would say it's very unlikely. As for provider- I have no idea but the best thing would be to ask when you get there and if it's possible you should be able to pick up the appropriate SIM in Mfuwe. In the Mfuwe area there is both internet (eg at Flatdogs) and mobile phone coverage.
Weather- beater is right about September, but you will definitely need that fleece earlier in the season. The earliest I've been in South Luangwa is late July this year and I was needing a fleece for early mornings and this year was a warm year. I'm sure you will need in for May and June.
Medical facilities- there is a volunteer doctor based at Flatdogs. The nearest hospital is a couple of hours drive from Mfuwe.