If I'm understanding correctly, an entire attribute (with all its levels) is to be removed in the UK sample. That means an entire row should disappear in the CBC question when showing to UK respondents; whereas US respondents see that additional row in the CBC questionnaire.
You could use conditional display or unverified perl IF to accomplish that. However, it leads to some interesting challenges if you want to try to do this as a single CBC exercise (and single analysis).
UK respondents would still have it represented in their data file that they saw levels for that conditional attribute (1s, 2s, 3s etc. would be recorded in the data for levels of the missing attribute). That means that the analysis would look at that design info for UK respondents and try to fit utilities to UK respondents as if they were reacting to levels of that missing attribute. This would introduce a tiny bit of random noise to the utilities for the other attributes. Plus, you'd have non-zero utilities in the utility file for that missing attribute based on draws from the population means (which included US respondents). The way to correctly deal with this would be to post-process the data file to make it look like the UK respondents saw levels coded as 0 (missing) for the missing attribute, not 1, 2, 3, etc.
Creating two CBC exercises in the same study would allow you to delete the attribute from the design for the UK people. That would mean you'd need to estimate utilities separately for the two populations and have two different simulators for the populations. There would be two completely different experimental designs.
But, you could take the second route above and with a bit of fancy data processing on the .HBU file, you could combine the two groups of respondents and add utilities of 0 for the missing levels of the missing attribute for the UK respondents. This allows you to keep all the respondents together in the same market simulator.
Regarding the standard errors, the standard errors for the non-conditional attribute are essentially unaffected. But, the standard errors for the missing attribute would in reality be larger and achieve the precision that only the US respondents could support. The UK-based respondents (if the data are processed correctly and the levels for the missing attribute are set to 0 in the experimental design for the UK respondents) do not contribute any information to making the conditional attribute any more precise than you would get from the US-only respondents.
Those are my thoughts.