Yes - 1080p has the potential to look worse on a 4K TV than a 1080p TV due to the upscaling required, however many good quality TVs do a really good job of upscaling and the difference isn't as noticeable as you might think. Bear in mind that people also often take the opportunity to increase the screen size when upgrading though, which won't help. One thing I found jumping from a 46" 1080p plasma to a 55" 4K OLED is that the OLED really shows up any deficiencies in the quality of the source material, with SD (and even some low-bitrate, streamed HD) looking poor and definitely worse than it used to on the plasma.
Your best option is to check reviews of the TV you're interested in - most good reviews (such as those on AV Forums) will mention how a TV performs with different sources at different resolutions and should comment on the upscaling. All 4K TVs will upscale lower resolution content to 4K.
The issue these days is that all mid-size to large TVs are 4K (here in the UK at least), so anyone wanted to upgrade won't have a choice ! The choice you do have, though, is to choose a set which upscales well.