Two suggestions.
If by 'pushing' you mean seating or wriggling the tube in the sockets to make it make contact, the likely cause is that the contact springs in the sockets are loose or flattened and not pressing on the tube pins. See solution #1.
If what is happening is that the tube starts when you handle it, regardless of whether you are causing it to make better contact, then see solution #2.
#1 So long as the plastic is not broken, the contact between the sockets and the tube pins can probably be improved. By careful wriggling, locate the culprit if possible; one end might be OK and the other intermittent. Then switch off,
isolate the circuit and turn off the main switch of the consumer unit to get both poles isolated. If possible, prove that the fitting is dead. Remove the tubes and look into the sockets through the holes where the tube pins engage. You will see metal contact springs either side of the holes that should press against the pins. Use a pointed tool to scrape off any deposits if the pin has been arcing against the contact spring, then gently hook the tool behind the contact and tease it out towards the centreline of the hole, only by a millimetre or so. Check and clean the tube pins, refit, re-energise and and re-test.
As using the tool to manipulate the contacts will put you in solid contact with the circuit, I cannot stress too highly the importance of properly and securely isolating the circuit. If you have removable fuses, remove the appropriate fuse and keep it in your pocket.
#2 Question: Do the tubes always flash as they start up (plink, plink, fizzz) or do they just come on and/or ramp up to full brightness? If the latter, this uses an autotransformer or semi-resonant ballast not a switchstart choke. They were popular in the 1960s but are more demanding of the correct type of tube. I can see earthing contacts pressing on the tube caps and foils around the ends of one of the tubes, which definitely suggests one of the two starterless ballast types (also the lack of access to the gear to change the starter). These are supposed to be used with special striped tubes like this:
Metal-Stripe Tubes (TLM) - https://www.firstlightdirect.com/metal-stripe-tubes-tlm
Plain tubes might work but might give trouble starting. The tubes look like T12s, which will probably work better than modern T8s. But if the contacts are good and you still have trouble with starting, try striped T12 tubes. If the stripe only reaches one end, that end goes in the fixed socket, not the retractable one.
If you can't identify the problem, post video of it starting.
