I am no longer in UK, but think I can speak to this.
In my UK experience, yes, it will cost you extra. There will be a scale, according to the value of any instrument you wish to insure. There's the rub! So if you have a piano which you bought for 2o,000 pounds you will pay the premium on that value and that will be similar to an equivalent value violin. Consider what cover you will get. You aren't going to take it out of the house, and nobody is going to steal it. Anything that happens as a result of playing it will not be covered. So really you are getting cover for accidental damage or fire. Both of those are fairly unlikely scenarios:
In the case of accidental damage that is most likely to happen if you decide to move it. How likely are you to want to do that? If you suddenly want it in the ballroom instead of the music room, get professional movers to do it (probably a good idea anyway) and they will have insurance. For knocks, scratches etc. get a good padded cover.
Fire? If your house burns down, well you will have plenty of other worries. I do not treat this subject lightly. In my house, here, fire is a genuine risk. So I have a fire hose in the boiler room, dry powder extinguishers in the hall, landing and kitchen plus a fire blanket in the latter.
I am not offering financial advice, merely explaining my mind set. By all means PM me should you so wish.
