The United Kingdom of Great Britain and Northern Ireland does not have Administrative 'Counties' as a ubiquitous level 2 administrative unit. Some places are Unitary Authorities (54 regions), others are Metropolitan Boroughs within Metropolitan Counties (36), so just traditional counties will have numerous gaps. What you need is Administrative level 2 boundaries and not counties. Admin Level 0 is the entire UK in one geography, Level 1 are the countries of Scotland (Alba), Wales (Cymru), England, & Northern Ireland (Ulster) in four geographies, and Admin Level 2 are the counties and the other geographies I list above with complete UK coverage in a consistent manner and detailed outlines.
1) This is what you get from GADM and I think it is just what you want. Level 2. Just go to the link and select the UK and select admins.
2) Download piecemeal from OS.
3) Obtain from EU NUTS.
Essentially, if you take number 1 then all the work is done for you and I suspect all 3 sources have the same data originator.