Yeah pretty confusing but to be fair the government have quite a bit on their plate atm! As I understand it, the furloughed worker (leave of absence) is managed by the employer who then claims the 80% back from the government through HMRC who are frantically trying to set up a system.
In one of my career reincarnations I worked with HMRC developing their Real Time Information (RTI) PAYE systems. It was an absolute shambles, took them years to set-up and after going live it was nothing but headaches for everyone concerned, for several more years after that.
All indications are that the similar system for VAT "Making Tax Digital" has been exactly the same. With several stages of the rollout postponed by months, if not years.
In theory, they should be able to extract the data from the RTI system to calculate 80% of employee's net pay. Then total it up and generate BACS files for the payments to the employer. Where it gets complicated is employer's that have some employee's on furlough and others in key areas that are continuing to work. There could also be a huge opportunity for fraud by companies that have a large turnover of staff on a regular basis. Plus, I doubt they would update the RTI system to do it automatically (that big a change in a short period of time, too much opportunity for huge screw ups).
In addition, the latest announcements of payments for the self employed will be much more difficult to verify and calculate. There's no hope of the HMRC developing a reliable new automated system to cover that within the next month or two. (Although I bet they have had plenty private companies putting forward a tender for the job, making all sorts of huge promises.)
I wouldn't be surprised if it'll become a matter of getting the staff in to process claims manually, which still sounds practically impossible. Worse with the various restrictions produced by isolation measures. However they do it, it's going to be a huge operation for another government department that has been cut to the bone by austerity measures.