Indeed, I have definitely seen an increase in "convenience" or "time" driven offerings for private healthcare - usually focused around opening hours or "avoiding waiting".
Some seemed to be trying to deliver NHS services (video/remote consultations available by "gig-working" GPs in evenings etc.), but others seemed to be increasingly trying to provide testing or other preventative/diagnostic services (blood tests, 7-day health check-ups booked same-day etc.)
I guess to some extent there likely will (among high earning, time-strapped immigrant workers) be an increased appetite to pay for healthcare directly, but for those who are aware of their immigration NHS access charge, they might well come to adopt the "I already pay for this" view. There's also a scalability question beyond London, where typical incomes mean the only people using any kind of private healthcare will be those with company-paid schemes.
There are some interesting ethical quandaries around what the NHS could do to improve its service (which could potentially hurt private sector rivals) - from some people's perspectives, the NHS could allow people paying privately to get things faster (without compromising on clinical need based priority) if it meant the service was better funded through revenues from those private patients going towards cutting wait times more generally. The British pastime of queueing could negatively impact on perceptions of anything like this though (even though you can already pay a little extra for an expedited next-day passport renewal etc.)
I know some hospital trusts have setups with private wards where NHS staff deliver private care (without bypassing priority by clinical need), and revenues from this are reinvested into regular services.