Steve, I like the work of Loll and Ambjorn too.
the most recent update on what they are doing, that is available online, was given by Ambjorn at the June 2007 Loops conference.
She gave an invited talk at the GRG 18 conference in Sydney in July, so maybe they are taking turns and it was Ambjorn's turn in June. they have been collaborating since around 1998, if I remember correctly.
If you are interested in their approach you should probably listen to Ambjorn's talk.
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one thing that you will find out is that when the application is to COSMOLOGY then Bojowald and LQC have a big advantage because they can calculate numbers.
they can use both analytical tools AND numerical simulation
more new researchers have swarmed into LQC. there is more to do. many cases to study. things to calculate, in the traditional fashion of physicists.
also LQC has been consistently predicting a prior contracting phase, which is very interesting
it also has a brief phase of exponentially accelerating expansion that happens naturally without the need to put in an inflaton field----so it offers the chance of developing an alternative to conventional inflation scenarios
(alternative ways to resolve the horizon problem, flatness, structure formation...)
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I agree that Loll's approach is extremely interesting and I urge you to listen to Ambjorn's talk, to get up to date on latest news about it.
But just in terms of applications to cosmology, there is currently much more activity in LQC
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Loops 07 program
http://www.matmor.unam.mx/eventos/loops07/program.html
plenary (invited) talks page (Ambjorn, Bojowald, Ashtekar...here)
http://www.matmor.unam.mx/eventos/loops07/plen_abs.html
contributed talks page
http://www.matmor.unam.mx/eventos/loops07/cont_abs.html
There is a chance that the approach being developed by a postdoc in Loll's group at Utrecht, named Dan ORITI, may prove to be a framework in which Causal Dynamical Triangulations can be calculated analytically. Oriti gave a plenary talk at Loops '07 and mentioned this possibility.
I have to go. back later. this is interesting to think about----implications of QG in cosmology. glad you brought it up