In the UK, are dogs allows to travel first class?

Yes, they are!

When you travel with Virgin Trains, you can take up to two domestic pets with you for free. Every additional pet after that is £5 more, but the cost of travelling with your pet - priceless!

Cats, birds and other small domestic animals are welcome to travel with you on Virgin trains.

It would be wonderful if we could have the same opportunities here in Australia!