I provided my visa vignette when I came over in 2010 on my work visa. No one ever asked to see subsequent BRP's or anything else - ever.
Sirius told me in the past that the NHS system and Home Office system "talk" to each other and the HO updates would have passed automatically to the NHS.
The changes to the NHS came in from April 2015, under the Immgiration Act 2014. You arrived in in 2010 on an Intra-Company Transfer visa, when UK law at the time said that everyone lawfully in the UK could have bill free NHS.
For the Tier 2 (Intra-Company Transfer) visas from April 2010, the UK no longer allowed these to settle in the UK (have ILR). A 12 month cooling off period outside the UK was brought in if they wanted to return to the UK on Tier 2 General visa (a visa does lead to ILR). There was an exception for those who applied for their ICT visa between April 2010 and 2011 if they wanted a Tier 2 General (lead to ILR) but not much of one as they had to find a work sponsor and pass the RLMT. And I believe those on the Intra-Company Transfer visa did not pay the IHS until April 2016?
From 2015, UKVI have used a banner system on the NHS SPINE system on the patients name, so that the NHS can know if the person is trying to use the NHS without paying. Green banner and red banner. No banner, the NHS staff can phone UKVI to see what their status is in the UK.
So... from 2016, when a Tier 2 (Intra-Company Transfer) expires, UKVI will change the banner on their details on the NHS SPINE system to red, to show they must now pay if they overstay. But it will remain green if they put in an in time application to switch (in country) to FLR(M) and stay green if that visa is approved, until the end of that FLR(M).
I don't know your timeline for when you married a Brit and applied to switch (in country) to FLR(M), but that is what happened from the dates above.
For those that overstay, UKVI requested their details from the NHS. It was something like 5,000 illegals were caught by UKVI this way in the first 10 months of 2016.
It doesn't work very well for those in the UK on EU rules. e.g. a non-EEA citizen could turn up with an EU "RC" but that becomes invalid if their EEA citizen sponsor has stopped being a "qualified person". The NHS has no way of knowing that they must be billed unless the holder tells them and the same with employers if they have turned up at work. The UK did bring in a laws so that illegally working is now a criminal offence for the person doing it, but that Act arrived at the same time as the UK voted to leave the EU, and those that were not lawfully in the UK then rushed to make sure they were.