So sorry for your loss. I have no idea how things work in Great Britain, but we live in Bremen, a city in Germany. Here, you are allowed to bury small pets in your garden, but we don't have one and burying them in a public place is illegal. We could have taken Basil's body to the vet and I think he would have been made him into animal meal. But we learned that the animal shelter here has a graveyard for pets. We had them bury Basil there. It cost 25 €, but that was really worth it, in my opionion. You might want to look online or ask the cet if they know of any pet graveyards. Here, they're expensive if you want a proper grave, but an anonymous one like Basil's was affordable.
Unless perhaps you wanted to go out to the countryside somewhere and bury her there, I'm sure whatever the rules are you'd get away with it, I have a friend who buries all her animals in Milport.
I was a bit of a wreck this morning and just didn't know where to turn so a big thank you all very much for your kind words and advice. It has been very much appreciated. I liked the sound of burial and growing a tree in my parents garden but my son didn't want his wee baby buried and there was no way I was taking her to the council refuse site. I eventually took her to the Pet Crematorium in Larkhall this morning where a very sympathetic gentleman stayed open until I arrived, explained all my options and didn't make me feel too embarrassed to be weeping.
Nicola, thanks for the recommendation re Hyndland Vets, they are just doors down from the pet shop I use. I'll use them in future. I wasn't that impressed with the vet that I saw recently.
Wiebke, I am so glad that you posted your link. My son doesn't want another guinea pig because of the heartache (it was his first pet) but I don't want our bereaved wee furry, aerie faerie, pinning away. I will have a look and see if there are any rescue homes near me who will let her choose a companion in her own time.