60 years is a bad idea if you don't extend. If you needed to sell, you'd struggle.
However, once you've owned it for over 2 years, you have a statutory right to extend by another 90 (unless the owner is the crown and a few rare exceptions, but the crown agree to abide by it). There is an online calculator you can use to calculate an idea of what the premium will be.
In a nutshell the premium is made up of 3 parts and if the lease is over 80 years, you only pay 2 of them - another reason why banks usually only lend on 80+.
You can be sure that the price will reflect the short lease, so for example it's not really a £100,000 flat, it's a £120,000 flat.
You can find out who the landlord is for a £4 land registry search.
If the landlord cannot be traced, there are statutory procedures to extend when you have a missing freeholder.
So in conclusion: YANBU, but expect to pay a fair amount to extend and for it to take time and be a bit of hassle. All do-able but be prepared :)
The lease.org website referred to above is essential reading.