How long for a timid rescue dog to show their personality?

I have had my rescue dog for 3 months now. She is 2 years old. She is confident with other dogs and on walks. She continues to be fairly timid in the house and garden. She doesn’t appear to have any personality at all. In the house she eats and sleeps and nothing else. Doesn’t come for affection or attention and doesn’t want to play. She doesn’t interact at all. Is this normal for a rescue dog? How long before we see the real dog?

Normal...?...it depends of her past experiences or lack of it. But I would not worry about it.
Sometimes it can take looooooong time and I'm sure there is changes in her behaviour/attitude that might not even show any detectible signs to you or are that minor that they are not obvious. All those will amount until one day she will come out of her 'shell' and you will take notice of it.
All my dogs have shown their pleasure for thing they like or when they feel happy/content by doing quiet sigh. So I have used the same impression to show my appreciation or liking to their soft actions towards me same way....like talking their language.
Even the pups soon learned what it means and any little lick of tongue or leaning against me was responded with sigh from me. If your dog don't mind you showing little bit affection...maybe just sitting quietly next to her and resting you hand on her back for the moment. Sigh for a sign that you like to be like that and that you feel content....and that's it...nothing more than just you two there in that moment. Then after while move away without any other 'fuss' and carry on with what ever you need to do.
If you carry on doing something like that every so often, she will learn that is something you like to do with her and there is nothing she needs to worry about it.
When that day comes that she might move towards you for more attention or touching, reward it with a sigh again and maybe with some soft talk/word...'looovely girl'.
There is some personality in her, but it might not be what you expecting to see...it might be very gentle and subtle so it has to be couched and 'trained' to come out..

That’s really helpful, thank you. All my other dogs have been very affectionate, craving the love they’d never had. With this one it’s not really like having a dog at all. Do you think we’re talking weeks or months to show any character?

Possibly months. But some dogs are naturally not particularly demonstrative - mine, if he's feeling particularly affectionate, will rest his head against my leg. There's none of this undignified spaniel-type full body contact with tongues! So make sure you're not missing her true personality because you're expecting big gestures. Maybe just soulfully looking into your eyes is enough for her.

Queen Victoria said:

That’s really helpful, thank you. All my other dogs have been very affectionate, craving the love they’d never had. With this one it’s not really like having a dog at all. Do you think we’re talking weeks or months to show any character?

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I don't think there is really any way of knowing the time scale or if the 'character' you are after will ever even happen. Her character just might turn out something very different you have experienced in past.

and that reminds me of our Brandy. She was a 'second hand' dog. She wasn't affectionate dog what so ever...more of 'no nonsense' kind of girl...or 'strictly business', always either 'working' or just enjoying her own company...almost 'regal' that she would just be on her 'thrown'
She did tolerate us when we wanted to show our affections to her and she would sleep next to me in very human-like way..head on the pillow and all that.
It wasn't until the very last moment on the vet's table when she gave a lick to my husband..like saying final 'thank you' before the injection would take a hold that she would make any kind of move to return the affection.
She was my girl through and through so when hubby received that lick, it did break his heart...it was looong to wait that moment and as sad as it was and as late did it come...it was worth of waiting for.

With all our past 'second hand dogs' I never expected to see the true character to show up until at least the first year was through. Even as some may have been 'characters' straight away, they still change and develop a lot as they get used to us and our ways.

Just sit tight and enjoy 'the ride', you never know what she bring along...she might turn out to me more of a 'cool girl' but even then, she will find and have her own 'funny ways'...

This might actually be her character. Some breeds/types are not overtly affectionate or interactive. But whether she is/will be or not, you haven't had her very long, and it will take as long as it takes. Less is more with many dogs, and the more their people are desperate to snuggle them, the less they can cope. Being polite but not tactile is the way with this kind of dog. Let her set the pace.

Finsky, that was a very moving tale about your Brandy.

All of the above. Our latest rescue has taken a very long time to relax, to trust to dare to play with us. I'm still careful not to set him back. Too much attention from our side and he's uncomfortable, licks his lips and wants to retreat. But then you think: remember what he was like at first and the change in him is huge. The rescue lurcher before that expected to drape himself around your neck in the evening- and he'd come from a shouty alcoholic who forgot to feed him. Go figure.

Thank you so much for your help and advice; which I have read carefully and digested. I think the main point from you all is : be patient!