Yes, travel insurance is worth having for a trip through Manchester, though the reasons look different here than for a remote or tropical destination: this is a large UK city with well-developed hospitals, and the real value of cover is less about access to care and more about disruption, cost, and what happens if your plans change partway through the trip.
Winter is the main honest risk worth naming. Manchester Airport, one of the UK's busiest, sees its share of weather-related delays and cancellations between roughly November and February - fog, snow, and storms off the Atlantic all affect schedules here more than in summer - so trip-delay and missed-connection cover is genuinely useful if you are flying in or out during that stretch, whether you are starting or ending a wider UK trip.
Many visitors use Manchester as a base for day trips into the Peak District, the national park just to the city's east, for hiking and hill walking across moorland terrain that can turn from calm to genuinely testing weather within a single afternoon. Falls and exposure incidents on the Peak District's higher ground are a real, if modest, source of claims, so if hiking is part of your plan, confirm your policy covers hill walking rather than just general sightseeing around the city itself.
On the medical side itself, the honest point is that Manchester's hospitals are capable and accessible, so the bigger financial risk for many travelers is the cost of any treatment as a visitor rather than its availability, plus the value of cancellation cover if illness disrupts your trip before you even fly out. Check whether your existing health cover extends to the UK and specifically to treatment as a visitor rather than a resident, since the answer varies by home country and by plan, and general assumptions here are often wrong.