Dogs : As a general rule well behaved dogs are normally allowed on bus services at driver's discretion.

Luggage : As a general rule items which can be carried by one person are permitted on buses, at driver's discretion, provided space is available.
If it meets personal luggage allowance for UK rail travel you should be OK. Note that wheelchair users often have priority.

Bicycles (non-folded) : In most instances these are not permitted on buses.
See: Taking your bicycle on a bus.

Bicycles (folded) : Provided it meets UK rail travel definition of a folded bicycle (not all do!) then typically treated as luggage.
Some operators may require the bicycle to be in its bag - so might be worth getting a bag just in case its needed.

Wheelchairs : Many buses these days have space for a person in a wheelchair.
These often allow powered wheelchairs / scooters. See - Rica - consumer research charity.

Traveling on Smaller Buses!

Above comments are with your typical large town buses in mind but scheduled bus services are also operated using smaller vehicles such as minibuses and taxi-buses. In these cases it might be best to check with the operator first.
The 'Useful Leaflets' button on our departure times page often provides a link to the operator webpage or contact details.