My personal belief is that most markets in the West are increasingly polarising to very high-end, expensive & luxury products or mass-produced, lowest-common denominator cheap products. These days, reasonably decent quality can be found at both ends of the scale; it's branding/heritage (to justify the effect of higher labour costs on end price) that determine whether you can consistently sell a product at the high-end or not.
Applying that to clothes, if you just want a reasonably nice, decently fitting suit, you won't go to Savile Row (or even off-Row); you'll buy something cheaper and get it tailored. A company like Suitsupply is an example of that niche. If you want a luxury branded product and but aren't particularly nerdy about the clothes, you'll buy from one of the big designers. Armani and Ralph Lauren are examples of that. These days, SR bespoke only appeals to those who want a very different, more traditional experience; those who have some appreciation for the history of tailored clothes and like the idea of getting a suit made the old-fashioned way.
Trouble is, until fairly recently, that was an ever-shrinking niche. It does seem like that internet interest in tailored menswear should have some effect on the size of the niche, but it's never going to be massive. Womenswear managed to deal with this problem by subsidising their Haute Couture through leverage of the brand/image created by Haute Couture into their RTW lines. That took a lot of advertising money & time, and they were lucky enough to make that shift at the right time, socially.
SR has bigger problems. It's left it very late to make that shift, and has frittered time & money away on terrible licensing deals that have diluted rather than enhanced their brands, and failed to modernise their core business model until recently. To try to now compete in the RTW arena on a large scale (e.g. Brioni, Kiton, RLPL, Zegna, etc) would require massive, massive investment and advertising budgets. I don't think they're likely to get that. They could try growing organically, but I don't think they'll be able to really compete adequately on that basis.
Instead, I think they're more likely remain niche luxury manufacturers, ticking over but not really growing. There's nothing particularly wrong with that; it's a legitimate approach. They'll continue to use their tourist image to sell accessories and some limited RTW to pad out their income. The problem is that, as I understand it, they don't own the freeholds to their properties and that creates a massive problem in terms of increasing rental costs over the coming years.
Ideally, one of two things will happen: either the SR houses will become a rich men's playthings competing for prestige (in the same way Premier League football clubs have become) and so they'll benefit from a lot of money and relatively low expectations around financial returns; or they'll all agree to move en masse to a less expensive part of town and use the undoubted blitz of publicity around such a massive move to cut prices and so open the bespoke market up to more people again. It would still be a very expensive product, but slightly less so, and would enable them to compete more effectively with designer RTW. This would be an insanely ballsy move; they'd lose the prestige of the Savile Row tag, and would need to rapidly establish their new road in popular culture's mind. It would take an unprecedented degree of co-ordination and with the risk involved, frankly won't happen.
Realistically, neither of the above options will happen but the firms will continue to muddle along, organically trying to increase sales of accessories and limited RTW lines, as well as raising prices on their bespoke offerings, hopefully just about keeping prices at a level sufficient to pay the rent, and trusting that as the world generally gets richer, enough global elite tourists will still want a suit from Savile Row. IMO that's actually the most risky approach, given property inflation in London, but it does have the benefit of inertia behind it.