@Potionqueen - No, they don't count you as one of the 13,000 (assuming you are not being paid).
The figures come from the NHS workforce statistics which are extracted from the NHS payroll system. They should therefore be an accurate reflection of the number of staff being paid by the NHS.
As of July (the most recent figures available) there were 301,836.36 FTE nurses and health visitors. The equivalent figure for July 2019 was 288,404.96. So the number of FTEs has gone up by over 13,400.
If, instead of looking at FTEs, we look at the actual number of nurses and health visitors, that has gone up from 323,810 to 337,931 over the same period, an increase of over 14,100.
Both figures represent growth of around 4.5%. So, whilst the raw number sounds impressive, it doesn't mean your local hospital has a flood of new staff. For example, the nearest hospital to me employs around 350 nurses so, if it is in line with the national average, it would now have 15 or 16 more nurses than it did this time last year, which works out at less than 1 per ward.
Re Matt Hancock, it is claimed on social media that, when he claimed 7,810 more doctors working in the NHS, this included unqualified medical students and that, if you removed these, the number of doctors has gone down. This claim is false.
The figure included no unqualified medical students at all. It did, however, include 1,948 doctors who are either on their first year of foundation training (which means they are fully qualified although, in normal circumstances, they would not be able to register with GMC until they had completed their first year) or who are final year graduates who have been allowed to fast-track into foundation training due to the epidemic (again, fully qualified). This was acknowledged in the government's press release. Even if we exclude all these doctors, the number has still gone up by 5,862 - a 6% increase.
Boris has certainly quoted some dodgy statistics in the past as have many other politicians of all political colours, but on this occasion he is correct.