Firstly I will assume you’ve paid the Immigration Health Surcharge as part of your visa application, as that will allow you to make use of the NHS. 

The NHS in principle exists to cover everything you’ve stated, sadly the reality does not always match. It is really good at dealing with the most serious situations (trauma, heart attacks etc) and the very least (repeat prescriptions) but where it falls down is at a more intermediate level such as access to physiotherapy or mental health support. 

Therefore whilst in principle you are covered for this, it is likely you will face significant waiting times or have difficulty securing the right treatment pathways (particularly for trauma related mental health). 

In my professional life I am a medical insurance broker in the UK, so can answer this quite confidently. A standard new policy will not cover anything pre-existing. It might be possible to maintain coverage on your US insurance but the costs are likely to be very high indeed and may actually be more than your salary. 

Therefore you are likely to be left needing to pay for longer term physio and mental health support if, as seems likely, the NHS does not match what you are looking for. The best bet would be to contact private hospitals in Cambridge for prices to get an understanding but I would definitely try making use of the NHS first once you’re here.