I’ve found various opinions on this, but I’d appreciate a fresh take.

We arrive on a Thursday evening around 21:00 to LHR Terminal 5. We depart out of Terminal 3 at 08:30. I believe (hope) our luggage will be checked all the way through and we will not have to collect it. We have booked a room at the Hilton Garden Inn. We are traveling with a young child and looking for the most convenient, fastest, and cost effective way (best balance) to get from Terminal 5 to the hotel in the evening and from the hotel to Terminal 3 in the morning.

Also, if our flight is at 08:30 and our luggage is checked already, how early do we realistically need to get back to the airport?

Thanks.