Disclaimer- this happened 2 years ago.
I had a WP in old passport. It expired, and so I applied for a new one. The old passport was returned to me, and I travelled extensively for 18 months with both passports (old with visa, and new that was valid). Never had an issue.
It's expensive to get the visa transferred. (although perhaps worth it if it's a ILR).
A nifty thing the UK does- if you apply for a new passport, and the old one is still valid, they will put X number extra months on the new one. I don't think the US does this.