I did this walk with a mate from church. Drive to Queanbeyan, then south along the Old Cooma Road, following the signs to Googong Dam. For the first walk, take the Burra Road, then London Bridge Road and park near the old shearing shed. For the second walk, backtrack to Googong Dam.
These walks are numbers 11 and 9 in Graham Barrow's book 30 Family Buskwalks in and around Canberra. The London Bridge circuit is rated as 4km walk distance, category easy and 1.5 hours walk time. The Black Wallaby Loop Track is rated as 3.5km walk distance, category easy and 1 hours walk time.
London Bridge is a striking geological feature and well worth the easy stroll to view it. We wandered on the tracks a little to the east, came back down Burra Creek, then strolled to view the Homestead.
Here's some interesting information copied from http://www.markbutz.com/Butz%201987%20Karst%20and%20caves%20in%20the%20Canberra%20area.pdf:
The London Bridge limestone is situated in NSW on Burra Creek, a tributary of the Queanbeyan River, about thirty kilometres south-east of Canberra City. The site was first described in the journal of Captain Mark Currie, RN, describing his 1823 journey of exploration, in which he discovered the Monaro Plains. On his return journey folklore has it that Currie was shown the Bridge by local Aborigines (P. Higginbotham, pers. comm.). He described it as ‘a natural bridge of one perfect Saxon arch, under which the water passed’ (Currie 1825), but he made no mention of any caves. The Bridge has been a well-known landmark since settlement began in the immediate area during the 1830s (Moore 1981) but access was very limited in recent decades due to the protectiveness of local landowners. The area received some notoriety through the ‘London Bridge mystery’ with the discovery in 1875 of human (supposedly European) bones in a small cave near Burra, leading to an inconclusive coronial inquest and rumours of murders committed by bushrangers (Moore 1981). Another story, seemingly unrelated, reports the discovery in 1874 of `a veritable catacomb’ containing `many hundreds of human bones and skulls, centuries old’. These were carried away by the bagful and pronounced by three surgeons, including the Coroner, to be `the skeletons of the Aborigines of former times’ (Brennan 1907). It has been widely believed that such Aboriginal cave burials are rare in the region (Flood 1980) but more recent analysis suggests that caves were used for burials (Spate, in prep.). The London Bridge has been widely investigated and well documented in scientific literature. The geological characteristics were recorded about seventy years ago (Carne and Jones 1919) describing a fairly pure white to dark grey fossiliferous limestone interspersed with thin shale bands and intruded by small granitic dykes. The outcrop was not considered to have great economic potential due to relative isolation and, as was stated: ‘it would, indeed, be vandalism to interfere with so remarkable a feature as London Bridge’ (Mahony and Taylor 1913). The limestone’s rich fossil fauna of brachiopods, corals, crinoids and trilobites and its relationship to other outcrops to the north and south have been described (Veevers 1953). The closest related outcrop, near the old ‘London Bridge’ homestead, has no caves but does have many fossils, an interesting metamorphic occurrence (Richardson 1979) and, therefore, some educational value. The geomorphic features of the site have been interpreted as a natural bridge and caves formed from complex repeated stream self-capture by karst action in a meander spur (Jennings et al. 1976); it is a classical example cited in international texts on karst geomorphology (e.g. Jennings 1985). The arch is thirty-four metres long, twelve to fifteen metres wide at water level, and about five metres high above normal summer water level. Six additional entrances are known, relating to two minor caves of four to five metres length and the more substantial Burra and Douglas Caves of about thirty-five and forty metres length respectively (Nicoll and Brush 1976a). For some years speleologists and geomorphologists, led by the late Joe Jennings, made representations to Government to establish London Bridge as a geological monument in recognition of its outstanding scientific interest and natural beauty and to open it to the public as an educational and recreational area (Jennings et al 1976). The area’s scientific and educational value was further reinforced by the recording of significant bone deposits in a number of the caves including remains of mammal species now locally extinct (Hope 1976). The sediment accumulation in the Douglas Cave has considerable potential to yield sub-fossil material. It is likely that this is the ‘very spacious cave, which bore traces of having been used in early times by Aborigines’ (Brennan 1907) and it still may contain evidence of Aboriginal occupation. No excavation has yet been attempted but careful stratigraphic investigation is well warranted (JH Hope, pers. comm.). In 1975 the Commonwealth acquired land for the protection of the foreshores of the newest element of the Canberra-Queanbeyan water supply, Googong Reservoir, and in the process acquired the London Bridge. For some time it was feared that this karst too, would be drowned. That these fears proved groundless was due far more to the Bridge’s elevation above sea level than to the sensibilities of planning engineers. The environmental impact statement (EIS) for the project recognised only the tourism value of the feature and assessed only the likely physical impact of flood waters on the rock itself. Other karst values were omitted completely, despite several submissions on the need to save the Bridge (NCDC and Dow 1973). Such shortcomings could perhaps be excused by the fact that this was the first EIS to be prepared under the Commonwealth Environmental Protection (Impact of Proposals) Act 1974 (Dalgarno and Minty 1983). In 1982 the southern half of the Googong Foreshores Reserve was opened to the public, thus affording to many local residents and visitors their first view of the London Bridge. Access was made available by a walking track, with the closest vehicular access at a distance of less than a kilometre. The formation has been nominated to the Register of the National Estate (AHC data file), and in that sense might be seen as a geological monument. While the aspirations of speleologists have been partially met by the provision of better access and nearby recreational facilities, and by National Estate recognition, the educational effort is at best a fledgling one. A general information leaflet for the Reserve includes a summary of the history and mode of formation of the Bridge. A number of the dates and inferences are inaccurate (AP Spate, pers. comm.) but this is being remedied. The leaflet associates the Bridge with the nearby ‘London Bridge’ homestead - a feature of major historical and architectural interest (Philip Cox 1983) which is listed on both the Register of the National Estate (AHC data file) and the Register of Classified Places (National Trust 1982). The Service has been offering guided walks to the homestead and Bridge for some years and these have proved very popular (P. Higginbotham, pers. comm.).
The Black Wallaby Loop Track provides great views down to the Queanbeyan River cascades and over Googong Dam. We also walked across the dam wall.