Can anyone help with an alternative to 0844 811 1191 please?

Have a look on the database, there is a number there. Have you tried that one?

In the database there were two entries for Icelandair, one verified and one unverified.

The verified one gave 0870 787 4020, 0870 787 4044, 0845 758 1111 and alternative 020 7874 1000. The two 0870 numbers are now not in service. The 0845 number and 020 number go through to the same recorded message and automated menu.

The unverified gave the 0844 number quoted in the OP and the same 020 number.

I haven't tried the 0844 number, but I think that it's likely that it goes through to the same menu. I have therefore removed the unverified entry and amended the verified one with the 0844 number (with the 0845 number in there as well).

The UK contact page gives the 0844 number and fax 020 7874 1001 which is the number immediately after the alternative we have.