Note for the helpful but unfamiliar with AAISP: an AAISP supplied router is highly likely not configured to use NAT and RFC1918 addresses. They do real routing wherever possible.
Not sure this is the case for their home packages -- pretty certain they are configured for NAT/ RFC1918 as they don't forward public subnets to all their customers by default. Otherwise Joe X would have a pretty hard time getting anything done (and they do market to him to an extent).
Their various offerings are excellent and I've been using/recommending them to businesses for donkeys.
Think the OP needs to:
1. Check with support whether the ADSL router is doing NAT/DHCP (call 033 33 400 999)
2. If not, get them to set it up (they'll know the customers WAN address and login as they're all static at A&A)
3. Disable DHCP on the Linksys, and put it on 192.168.1.0/24
4. Look into getting some address space from A&A (they provide public subnets for free*) if they want to do away with NAT altogether (not sure how this will work when OP is roaming on others' networks with smartphone, laptop, etc -- I use a mix of NAT'ed/ properly routed subnets behind m0n0wall/FireBrick or similar)
* Presumably because they want to get businesses away from NATing/ <spits>port forwarding</spits> and they understand that IPv6 will do away with all this nonsense anyway (ie, charging for IP address space) -- still, I've no doubt BT et al will continue to charge arms and legs even for IPv6 space.