score:3
As with many other question around Brexit, the answer is “we don't really know at this stage“. That said, a couple of observations and remarks to help you understand what to expect and what to pay attention to in the next couple of years:
The EHIC is part of all the EU/EEA rules designed to facilitate free movement (and especially free movement for persons), which is precisely what the UK government wants to call into question. So you can't assume it will stay and if the UK leaves the single market or fails to negotiate some sort of special relationship with the EU, British citizen will be kicked out of the system.
The E111 form was already an EU (or actually EEC) thing. Technically, it wasn't tied to “EU citizenship” (as that notion was created with the Maastricht treaty) but certainly to EU/EEC membership. To the extent that a so-called “hard Brexit” (clean break with all the existing treaties and regulations) seems increasingly likely, you can't assume we will revert to anything like that either.
There are some bilateral reciprocity agreements between EU and non-EU countries so it's not inconceivable that the UK would negotiate something like that either with the EU as a whole or with select countries (and Spain would likely be very high up on the list given the number of Brits living there). Such an agreement seems a lot less complex than trade negotiations and the entanglements with the rest of the EU rules or things like the WTO rules are limited so it could be relatively easy to do within the two-year timeframe defined by article 50.
It's also conceivable that Brits living in another EU country for a long time may qualify for some long-term or permanent residency rights or be grandfathered in any future system, either as part of a negotiated solution or even unilaterally. Some EU countries already have something like that for third-country nationals, without any relation to EU membership. Basically, the notion is that once you lived in a country for a long time under one of the usual residency regimes (work, spouse, etc.), your ties with that country are so strong that forcing you to leave the country without a serious reason (like having been found guilty of a crime) would be so disruptive that you are allowed to stay even if you don't meet the conditions anymore.
The details will depend on each specific situation and on the laws of the country in question but some British citizens may therefore be able to get healthcare on a par with citizens and other long-term residents (including subsidies or free healthcare if they are destitute) and therefore have something to fall back on once they are not covered by EU rules anymore. But that's unlikely to apply to newcomers or people who just moved into a specific country when the actual Brexit happens and given the specifics of your situation detailed in the comments, I doubt that you would be able to use that.