Hi Anees, welcome to TOC
1) as you are buying from a trader you have very good rights under the Consumer Rights Act 2015 should anything go wrong within 6 months (& the car is still within Toyota UK's warranty period anyway).
2) Toyota service intervals are 10,000 miles or 12 months whichever comes sooner. However, they generally allow leeway of plus 1000 miles or 1 month.
6 stamps should be 60,000 miles so it's probably due another especially as it is 10,000 since last - I probably would have made a service part of the purchase deal but hey ho ...
The fact that not all services are dealer should not impact on the warranty as such but may mean that you may not get goodwill for things outside normal warranty terms.