Just bought set of new tyres for my car -

have system pressure tested -and a blown headgasket doesnt always mix with oil

Yes

check the condition of the wheel bearings for failure or adjustment

Here's a link Johnny on how to replace gasket.

White smoke usually indicates coolant is getting into the combustion chamber, meaning that it most likely has a blown headgasket, or gaskets, if it's a V-6. The cost varies depending on your location, who does the repair, whether it's a V-6 or 4 cylinder. Ball park figure for a 4cylinder headgasket replacement would run somewhere between $800-$1200 ad an additional
$500 if it's a V-6.

might be your headgasket...are you losing coolant through the overflow? getting a hyrdocarbon test will show this very quickly...high carbons=blown headgasket

check fuel and spark.

Remove head from car, buy book, take to machine shop and have them resurface the head and inspect it from cracks or possible damage in valves or guides,To do the job You have to really done one before, call me 786 370 1812 Caesar s my name Ill be glad to help You with any questions I remember changing the spar plugs was removing the all top of a cover, very complicated technology from the late 80s

I would get a second opinion at another transmission shop. If the transmission has not been serviced properly I wouldn't be surprised that it is having problems.

Yes, driving around with a blown headgasket is BADDDD....

If you've never replaced a headgasket before then I would suggest that you not try it without proper supervision and help.
On a v8 when you replace one headgasket you replace the other as well. The gasket kit comes as a set of 2 headgaskets and all other gaskets and o-rings that will need to be replaced. You will need an array of tools to accomplish the mission - some hand tools and some specialty tools for dismantling the cylinder head.
I'm really not trying to scare you off but this is NO SMALL JOB. It will require patience and it will require tools and prior knowledge - so get the HAYNES RePAIR manual for the car and read up if you decide to go for it. I did the headgastets on my Acura Legend v6 - and what should have taken 30 hours start to finish took about 55.
Good Luck and Happy Motoring!!!!!!!