Dual nationality

The British government has never objected to its citizens holding other nationalities.
I, my late wife and two of her sons all hold dual Belgian/British nationality as do numerous users of this site.
My own son - with a French wife - and their 3 daughters all have dual French/British nationality.
As a Belgian national, you have to vote or prove that it was impractical for you to do so - you were away on holiday, for example.

Kasseistamper has answered most of this. Applying is a bureaucratic challenge, but not too bad. But you need to either have been paying social security here for 5 years with no break at all, or have been resident for 10 years.
As UK has now left the EU, you may be too late.

Just to answer your point on Spanish inheritance rights;

Having Belgian nationality will have no impact whatsoever on your inheritance rights in any country.

Now Brexit has started, is it correct that the British government does not object to its citizens holding other nationalities?

Yes

To complete all the formalities for a Belgian passport is it a mammoth task?

No

How long is the process?

A few months

Does anybody know what all the advantages and disadvantages are for having dual nationality (British and Belgian passports)?

There are no disadvantages, only considerable advantages

Does it have an impact on ones UK and Belgian social benefits, pensions and inheritance?

No

If I applied for a Belgian passport would I be obligated to vote in Belgium?

Yes, as a Belgian citizen you are obliged to vote

If I obtained a Belgian passport would this be to my advantage when claiming inheritance rights with a Spanish property?

No

As a dual British/Belgian citizen living in Spain with Spanish property, you have the right to write a will where who gets what is covered by British, Belgian or Spanish law. This is not affected by Brexit.

Spanish taxes apply to your estate, and possibly taxes from other countries where your assets are held. This is not affected by Brexit either.

Just a small remark to the length of the procedure:

once your application is considered full and eligible it will take from 4 up to 6 month, depending on the commune. Though, the tricky thing is to collect all the right documents that would be accepted by the commune you are registered with. Some communes are quite bitchy and might turn you back to collect some additional docs.
My advise is go the organisation "Objectief"- they provide quite a comprehensive information depending on the commune and your particular case;
then, go to the commune and check your eligibility and ask the list of the documents you need to provide. remember to get the list signed by the clerk, this will ensure they won't ask additional docs at a later stage, which might prolong the time.
and if you seriously think to start it, you should have started yesterday. :)