Firstly, PageLines Framework has not come to the end of life. DMS is a different product, operating in a different structure within your site(s). Thus there's no worry about being abandoned by us. That said, I cannot give you any idea how long Framework's lifespan will be. We're just recently ending PlatformPro's lifespan, more than 18 months since Framework was introduced, so I doubt we'll be in any rush to end Framework anytime soon.
Lifetime is defined as the lifespan of the product. As technology advances quickly, the lifespan of products may become shortened. Framework was flexible enough to last a while, so I'm confident that it will not come to end of life very fast.
Without Plus, unfortunately DMS will be a new purchase. Discount questions are asked via our Billing office at firstname.lastname@example.org.
DMS works (is visible) in all major, current browsers. It is only the visual management tools which work only in Chrome at present. This is the layout, and design, not the addition of posts and pages. So if your clients at NHS want to publish a post or new page from IE, they can still do so, since that's done in WordPress, not DMS. However, if they want to redesign the layout of the page too, or customize some aspects of its design, then Chrome will be essential. Given that, and without knowing exactly how this is organized, I'd say you're not stuffed.
I don't think you're "going down the pan" at all. Rather, I think your ideal tool of Framework will remain for some time, and you can continue using it, comfortably. I would however, recommend that in the fullness of time, you get the free version of DMS, experiment with and test it, as this will help you in future, should NHS upgrade systems that include contemporary browsers.
My own sites are remaining in Framework for the present, till I can find the time to convert. They're rather large sites (over 250,000) posts, so it's going to take a bit of time to convert everything. Time being the precious commodity I lack in abundance.