In summary, the next "13.00 Local Sidereal Time" for a UK observer is at 10:06pm and the star "Phecda" and the galaxy "M109" will be almost directly overhead. The centre of the Milky Way is tilted 30 degrees from the horizon and located to the southeast, 20 degrees below the horizon. There are no named stars within 10 degrees of the nadir. Moving north or south from the UK, there may be different stars near the nadir and zenith, but the centre of our galaxy's cluster cannot be pinpointed on the sky.
You can work it out yourself fairly easily. When the Sun passes through your local meridian (i.e. - noon), your local sidereal time is whatever right ascension the Sun is at. Right now, the Sun is at ~2h54m. To get to 13h00m, you therefore need to wait 13:00-2:54=10:06. That would mean that 13h00m passes overhead at 12:00pm + 10:06 = 10:06pm.
Though that presumes that your local noon, when the Sun is in the middle of the sky, is in fact 12:00pm! Depending on where you are in the timezone this may be incorrect, and you would have to add (or subtract) whatever is the difference.
A google for "sidereal time calculator" yields lots of hits. Here's the first: http://tycho.usno.navy.mil/sidereal.html [Broken]
According to Starry Night, the star "Phecda" and the galaxy "M109" are almost directly overhead at that time (given in the previous post) and the Milky Way is tilted roughly 30 degrees from the horizon from north to south (imagine a hula hoop around your waist, tilted up 30 degrees). The center of the Mikly Way is to the southeast, 20 degrees below the horizon. There is no named star within 10 degrees of the nadir.