What is true is that the NHS has been severely underfunded (if they want to continue to provide the same services) and that services have been cut as a result. https://fullfact.org/health/spending-english-nhs/
It may feel like there's less, because there's more people now, so the increase in services is offset by the increase in population. But that's also not a cut.
There's a fundamental design issue with any system that combines free healthcare with free movement. Neither party is interested in changing this situation, but Labour don't even pretend to try - they have adopted policies at their conference of essentially eliminating all immigration controls. It's hard to imagine what kind of budget a nationalised health service would require given no constraints on served population. Effectively it requires an infinite budget.
Health spending in every country has increased per capita (inflation adjusted), in the UK it has too. But in the UK, for 40 years, been way under that of countries like France, Germany, Switzerland, Denmark, Norway, and of course the U.S. The rate of increase hasn't increased either.
This isn't an issue of (typically working age) immigration - indeed that offsets the problem. It's an issue of an increasingly older population costing more per head. In the UK there's also an issue that people stay in hospital too long because of a failure in social care.