As described in the UK Track page, there's a fairly wide zone which offers pretty much equal viewing of the eclipse; anywhere within about 8 miles of the centerline should see a 2-minute total eclipse (or within a few seconds of that). Still, as some people might be interested in the exact centre track of the eclipse, I've done my own very rough plot of the track, and it's described here, accompanied by National Grid references. Although I've gone into a lot of detail, bear in mind that it may not be too accurate!
The first signs of an eclipse will be seen in Cornwall a few minutes before 10:00 BST (9:00 UT), when the Moon begins to move in front of the Sun. The partial eclipse will increase gradually until the total eclipse arrives.
The centre of the track of total eclipse hits the mainland at about Trewellard Zawn/The Avarack (SW 373 352). The front of the shadow will reach here just after 11:10 BST (10:10 UT), followed by the centre a minute later, and the Sun returning just after 11:12 BST (10:12 UT). The coast path there might be a good viewing point (by the same token, it might be thronged!); on a clear day, you'd see the shadow coming in over the sea, like an approaching storm; but rather quickly (over 2,000 miles an hour)!
Lower Boscaswell (just south of the centre line) falls under the shadow next, followed by The Carn, on Carn Downs (SW 411 347). Chûn Castle and Woon Gumpus Common are just south of the centre, and might make good places for a picnic. (They make these names up for the tourists. Really.) Alternatively, Watch Croft, at 252 metres above sea level, is a good high vantage point, just a kilometre north of the track.
The track moves on across an area full of public footpaths: but watch out for old mine shafts! Tredinnick, Boskednan, Ninnes and Carfury are all close to the track, which then passes to the south of Chysauster Settlement (with its public car park). Castle Gate is almost on the track just to the north, and nearby is Castle-an-Dinas, another high vantage point.
In terms of towns, Penzance to the south and Hayle to the north are not far off at all, but Crowlas is nearly on the centre. The butterflies at Fraddam are going to get a shock (about a mile north of centre), followed closely by Townshend which is pretty well in the middle. According to my reckoning, Trenwheal (SW 620 327) is going to be exactly in the centre of the eclipse at 11:12 BST (10:12 UT). That means that it will be in shadow from about 11:11 - 11:13 BST (10:11 - 10:13 UT).
Moving on, the woods of Porkellis Moor are on the track, but who wants to be under trees? The public information point on the A394 at SW 704 319 might be a good place to park, being right on the centre line. The track passes over Carvedras and Treverva (SW 756 315), followed by the southern edge of Falmouth; the sunbathers on Swanpool Beach are on the centre line as it reaches the sea (SW 803 311). Pendennis Point and Zone Point are just missed by the track, but not by much.
You might go farther east to Porthmellin Head, slightly north of the track, to say goodbye to the eclipse a few seconds later. Having said that, the area of totality is 100km wide; Dartmouth in Devon, although not on the centre of the track, will be seeing a total eclipse 2 minutes after Falmouth is in sunshine.
After this brief but spectacular pass of the total eclipse, Cornwall returns to a partial eclipse, which will decrease gradually until the Moon finally parts from the Sun shortly after 12:30 BST (11:30 UT).