The single biggest issue is that you can't control the queues at CBP or security screening. If it's a bad day there won't be time for a shower.
Queues for security screening depend on whether it's a day where they're using full body scanners for all passengers and doing shoe/belt removal as well as LAGs out of bags. Strike a day when they're using regular metal detectors with no shoe/belt removal and the queues move incredibly quickly.
On my last trip back from LHR in July there was no priority queue for transit passengers, but that was on a day where they weren't using full body scanners or enforcing show/belt removal so maybe with the fast moving queues they decided there was no need.