How to use an Irule switch to match HTTP::host and HTTP::uri?

I'd suggest putting a logging statement in your problematic match clause and verify what you are expecting to see in the log:

when HTTP_REQUEST { 
 switch -glob [string tolower [HTTP::host][HTTP::uri]] {
  "www.liveapp.customer.co.uk/app1" {
    pool CUST1-WEB-PROD-APP1-Pool 
    SSL::disable serverside
  } 
  "www.testapp.customer.co.uk/app1" {
    pool CUST1-WEB-PROD-APP1-test-Pool 
    SSL::disable serverside
  } 
  "www.liveapp.customer.co.uk/app2easypath" {
    pool CUST1-WEB-PROD-APP2-Pool 
    SSL::disable serverside 
    HTTP::path [string map { /app2easypath /realpath } [HTTP::path]]
  }
  "www.testapp.customer.co.uk/app2easypath" {
    pool CUST1-WEB-PROD-APP2-test-Pool 
    SSL::disable serverside 
    HTTP::path [string map { /app2easypath /realpath } [HTTP::path]]
    log local0. "Request to the server is [HTTP::host][HTTP::uri]"
  }
  "www.liveapp.customer.co.uk/" {
    pool CUST1-WEB-PROD-Default-Pool 
    SSL::disable serverside
  }
  "www.testapp.customer.co.uk/" {
    pool CUST1-WEB-PROD-Default-Pool 
    SSL::disable serverside
  }
  default { HTTP::redirect http://www.google.com} }
  }
}