Hey Penny....
..... so ..... coconut milk..... I went to try and find an example for you, and I was able to find that Countdown stores have this coconut milk:
https://shop.countdown.co.nz/?banner=www#url=/Shop/ProductDetails%3Fstockcode%3D793905%26name%3Dvitasoy-coconut-milk-original%26searchString%3Dcoconut%2BmilkIt is an example of what I'm talking about. Growing up, if you had said coconut milk to me I would have thought the stuff inside a coconut too. But, the above link should show you more of what I mean.
Your English breakfast tea should do well with an alternative milk though. We have had troubles trying to put other milks into herbal teas. It separates and is gross.... except for mint. But, real caffeinated tea that we've used will take other milks well. It then just comes down to what flavor you like.
If you are able to get some other milks to try, if there is a vanilla version, I'd go with getting that too. Cooking you can go plain, but drinking I'd go vanilla.
BTW, you can also make your own rice milk. We buy ours.... which really is only used on cereal or in cooking. But, for anything where you might seriously taste it, I did make rice milk once and it was the only time we all thought it was good straight. You can find directions on the internet. It is a lot of work ... but if you don't have 3 kids bugging you while you try and make it, it might not seem like such a hassle!