Let's suppose that, for one reason or another, the transistor never came about at all. How far can we take computing?
1936 - Zuse Z1 is a fully mechanical computing machine, using sliding parts for memory and linkages for logic gates.
1941 - Zuse Z3 uses electromechanical relays for logic, valves for memory.
1942 - A capacitor-based form of memory is patented.
1947 - development of magnetic drum memory.
1949 - Following a conference organised by John von Neumann architecture, EDSAC and CSIRAC are completed using valves for logic, ultrasonic 'echoes' in mercury lines for fast memory.
Arguably one of the most advanced computers that did not make use of silicon was BESK (1953-1966) using germanium diodes, vacuum tubes and magnetic drum memory. After this point, essentially all computers in our timeline were transistorised. Reliable, small, power-efficient transistors effectively killed any motive to further develop valves. As we are just now beginning to discover, tubes were not a technical dead end - in fact, they may offer some advantages over semiconductor technology in the future. A speculative alternate timeline runs:
1955 - vacuum tubes continue to be miniaturised. John Von Neumann suggests including multiple tubes into a single package with a common heater coil, inventing the integrated circuit.
1959 - The vacuum fluorescent display is developed by Phillips, setting the stage for metal-in-glass package development with small-scale features. A smaller line of specialised computing packages is developed alongside VFDs for commercial and military computers.
1959 - development of the cold-cathode tube dramatically improves power consumption, as unheated filaments become usable for calculation.
late 1960's - photoresist-based manufacturing becomes well-understood enough to make small features. Cost of computers begins to fall, as more valves can now be squeezed in to less space.
1977 - Discovery of scandium-doped cathodes, which give good valve performance even at comparatively low temperature, dramatically improves power efficiency of valves.
1970's - Vacuum-channel transistors become reliable enough to replace magnetic core memory in computer ram. Clock speeds increase dramatically, but it proves somewhat difficult to scale up the number of transistors due to the slow development cycle of sealing things into glass packages.
1980's - Nanofabrication technology, combined with improved cathode materials becomes good enough to create valves small enough to avoid the need for a vacuum entirely.
1990's - First consumer-grade computers. Uptake is limited by the manufacturing costs of these rather niche components, combined with limited component lifespan due to inevitable failure of the small features with regular use.
These microvalve-based computers would have an number of disadvantages compared to transistorised computers, mainly to do with power and failure modes. I imagine the jump to mobile devices would take much longer without silicon, but we will never know - given good motivation, technology has a habit of flowing around obstacles rather than being stopped by particular roadblocks. By 1950, the usefulness and need for computers was already considered self-evident by experts, and it is virtually certain that the redirecting the money and effort thrown at semiconductors in the '60s and '70s and to vacuum tubes would quite likely have resulted in equally impressive improvements in performance and reliability.