How might the issue of US imposed tariffs impact US relations with Britain?

Since no time frame is specified, I will speak on the present. If the US in 2016 were to impose a tariff on UK goods, it would be seen as a violation of the European Common Market and a general insult to the US-UK Special Relationship. It may also be a violation of MFN requirements at WTO level depending on the extent and height of the tariff.