I don't think there will be places where you have to have a contact to even apply or stand a chance.
I believe that would be illegal under rules governing public sector employment in the UK - nepotism just ain't allowed.
There are (rare) occasions when jobs are not as widely advertized as they should be - they just have to be advertized somewhere - but I have only ever seen this for temporary lectureships or post-docs and with the advent of jobs.ac.uk the tendency to bury jobs in the vacancies section of a local newspaper seems to have disappeared.
I have been involved in recruitment of several faculty members and have never seen 'who' writes the reference for a candidate have any effect on recruitment; some institutions (my Russell Group employer is one and I don't know if this is universal practice) only use references to check there is nothing untoward about a candidate
after interview. It is several years since I have a seen a candidate reference as part of the short-listing process.