If you were in the UK for the bulk of the year preceding and following the current tax year, then the treaty says all workdays are UK workdays.
If you don't invoke the treaty, then the US workdays are taxed in the US. The UK won't give you relief for those workdays, however, since you *could* have taken the treaty position but opted not to.
Many of those "workdays" might not really be workdays. A conference or educational trip to the states could be considered outside the scope of workdays. This is a position to discuss with your tax advisor.